Infinite Degrees of Speed: Marin Mersenne and the Debate Over Galileo's Law of Free Fall

Infinite Degrees of Speed: Marin Mersenne and the Debate over Galileo's Law of Free FallAuthor(s): Carla Rita PalmerinoSource: Early Science and Medicine, Vol. 4, No. 4 (1999), pp. 269-328Published by: BRILLStable URL: http://www.jstor.org/stable/4130144 .

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INFINITE DEGREES OF SPEED MARIN MERSENNE AND THE DEBATE OVER

GALILEO'S LAW OF FREE FALL

CARLA RITA PALMERINO University of Nijmegen*

Introduction

In 1634, Marin Mersenne published a short pamphlet entitled Traite des mouvemens, et de la cheute des corps pesans in which he presented the results he had obtained when measuring the natural acceleration of falling bodies. After years of stubborn adherence to the idea that free fall had to take place with uniform speed, the evidence of accurate experiments had finally persuaded the Minim to accept the validity of the law of acceleration which Galileo had proposed in his Dialogo sopra i massimi sistemi.' Measuring the times used up by a leaden ball falling from heights of 147, 108, and 48 feet, respectively, Mersenne had persuaded himself that "the speed of this ball increases in the duplicate ratio of the spaces that it

* Research for this article was made possible through the financial support of the Netherlands Organization for Scientific Research (NWO), grant 200-22-295. I wish to thank Peter Damerow, Peter Dear, Cees Leijenhorst, Sophie Roux, and the Editors of this Journal for their useful comments on earlier drafts of this paper.

NB: The abbreviations used in this article refer to the following texts: C.M.= Correspondance du P. Marin Mersenne, religieux minime, ed. C. De Waard,

R. Pintard, B. Rochot, A. Beaulieu, 17 vols. (Paris, 1945-1988). A.T.= Oeuvres de Descartes, ed. C. Adam and P. Tannery, 13 vols. (Paris, 1897-

1913). G.G.= Le Opere di Galileo Galilei (Edizione Nazionale), ed. A. Favaro, 20 vols.

(Florence, 1890-1909). P.G.= Petri Gassendi Opera Omnia, 6 vols. (Lyon, 1658).

In letters written between 1628 and 1631, Mersenne repeatedly expressed his conviction that bodies fell with uniform motion. Despite the fact that this hypothesis had been refuted by Cornier, (Cornier to Mersenne, 29 March 1628, C.M., 2: 52), Descartes (Descartes to Mersenne, 18 December 1629, A.T., 1: 94; C.M., 2: 344) and Beeckman (Beeckman to Mersenne, 30 April 1630, C.M., 2: 457), Mersenne continued to express his support for it in 1631 (Mersenne to Rey, 1 September 1631, C.M., 3: 188). Mersenne's position is summarized and criticized by Beeckman in Journal tenu par Isaac Beeckman de 1604 a' 1634, ed. C. De Waard, 4 vols. (The Hague, 1939-1953), 3: 181-182 (December 1630).

? Koninklijke Brill NV, Leiden, 1999 Early Science and Medicine 4, 4



270 CARLA RITA PALMERINO

traverses in falling, and hence that the spaces grow in the duplicate ratio of the times."2

But although this ratio of acceleration corresponded to Galileo's law, the numerical data presented in the Dialogo turned out to be

quite inaccurate. Mersenne therefore decided to publish a Traite so as to present his "experiences veritables."3

After instructing his readers in the art of calculating the time

employed by a falling body if the height is known, and of finding its initial height if the time of its fall is given, Mersenne devotes the last pages of his treatise to the possible causes responsible for the fall of bodies. Although he manages to think of three such

causes-"positive and real heaviness"; the pressure of the air on the falling object; and terrestrial attraction-he rejects them all for different reasons: the first appears to be incompatible with the

possibility of acceleration; the second implies that bodies of different weights will fall with different speeds; and the third entails that the degree of acceleration should be weaker at greater distances from the earth.4

With his tables of experimental results, Mersenne managed to correct, it is true, Galileo's approximate data. But he did not succeed in filling an essential lacuna of the Dialogo: "gravity," which in the Galilean vocabulary had remained an undefined term, was doomed to remain so even for Mersenne. His treatise ends in fact with the resolve to be content with a description of natural phenomena and to do without knowledge of the underlying causes and principles.

If one believes that free fall depends on the attraction or magnetic force of the earth, one can say that falling bodies keep a geometrical proportion, all the more as the activity of all sorts of natural agents diminishes with the

squares of spaces. But I have no doubt that one can invent various other reasons to explain the proportion maintained in the speed of free fall,

2 M. Mersenne, Traite des mouvemens, et de la cheute des corps pesans et de la proportion de leurs difffrentes vitesses (Paris, 1633), 2: "[...] la vitesse de ceste bale

s'augmente en raison doubl&e des espaces qu'elle fait en descendant, et

consequemment [...] les espaces croissent en raison doubl6e des temps." ' On the subject of Mersenne's mistrust of Galileo's experimental assertions, cf. A. Koyre, "An experiment in measurement," Proceedings of the American Philo-

sophical Society, 97 (1953), 222-237; P. Dear, Mersenne and the Learning of the Schools, (Ithaca, 1988), 136-137; Id., Discipline & Experience: the Mathematical Way in the Scientific Revolution (Chicago, 1995), 129-132. For an analysis of the Traite des mouvemens, cf. also P. Boutroux, "Le Pire Mersenne et Galil6e," Scientia 31 (1922), 279-290 and 347-360, esp. 347-350.

Cf. Mersenne, TraitS, 21-23.



INFINITE DEGREES OF SPEED 271

though it is maybe not less difficult to find its true cause than it is to demonstrate whether the earth is stable or mobile. That is why it is sufficient to explain the phenomena of nature, since the human spirit is not capable of possessing its causes and principles.5

Mersenne's scepticism, which in these lines still appears "mitigated" by his faith in the experimentally ascertained fact, was in the course of time to give way to a radical sense of pessimism in relation to the possibility of constructing an exact science of motion. The analysis of the fall of bodies (de casu gravium) offered in the Novarum observationum ... tomus III of 1647 betrays clearly Mersenne's recent conviction that the mathematical description of the phenomenon of free fall and the analysis of its physical causes cannot be separated. The Galilean law of odd numbers, which Mersenne presented in 1634 as the only rule in conformity with the observed phenomena, is depicted in 1647 as merely one of the

possible laws. And although Mersenne still admits that bodies

dropped from modest altitudes appear to confirm the law formulated by Galileo, he now insists that our ignorance of the true cause of acceleration does not allow us to eliminate any of the rival conceptions:

You see therefore that of these descents of bodies, which are commonly called heavy bodies, nothing deeper can be demonstrated as long as the principle, or the true and immediate cause, is unknown for which such or such bodies begin their way towards the center, and how much they are aided or impeded in this way by all other bodies they meet or which sur- round them.6

Peter Dear has formulated an interesting hypothesis to account for what he calls Mersenne's "change of perspective from pragmatic

5 Ibid., 24: "Si l'on tient que la cheute depend de l'attraction, ou de la vertu magnetique de la terre, I'on peut dire que les corps qui tombent gardent la proportion geometrique, d'autant que l'activit6 de toutes sortes d'agens naturels se diminfie en raison doubl6e des espaces: mais je ne doute pas que l'on ne puisse inventer plusieurs autres raisons de la proportion que gardent les cheutes dans leur vistesse, quoy qu'il ne soit peut estre pas moins difficile d'en trouver la vraye cause que de demonstrer si la terre est stable ou mobile. C'est pourquoy il suffit d'expliquer les phenomenes de la nature, puisque l'esprit humain n'est pas capable d'en posseder les causes et les principes."

6 M. Mersenne, Novarum observationum physico-mathematicarum ... tomus III, (Paris, 1647) 133: "Vides igitur de his casibus corporum, quae vulgo gravia dicuntur, nihil penitus demonstrari posse donec innotescat principium, seu vera et immediata causa ob quam versus centrum haec et illa corpora suum iter instituant, quantumque iuventur aut impediantur in toto itinere ab omnibus aliis corporibus occurrentibus aut circumstantibus."




adoption to principled rejection" of Galileo's law.' According to Dear, Mersenne's gradual estrangement from the original Galilean

position was probably influenced by his reading of three works, published between 1644 and 1646, in which the validity of the odd- number law was disputed. These were Descartes' Principia philosophiae (1644), which strongly emphasized "the dependence of free fall on the nature of underlying physical causes,'"8 and two

important treatises by Fabri and Baliani, both published in 1646, in which Galileo's interpretation of acceleration as a continuous

process was called into question.9 These works could not have failed to undermine Mersenne's original "ambition to push forward a mathematically certified kind of natural philosophy."'0

Now, it is undoubtedly true that Descartes, Fabri, and Baliani all tried to refute the Galilean law of acceleration on the basis of

physical considerations. But it is equally true that, at the same time, there was an author much appreciated by Mersenne who

pursued exactly the contrary aim, that is, to prove the validity of Galileo's law by physical means. This author was Pierre Gassendi, who between 1640 and 1646 wrote six Latin epistles in which he

attempted to provide the Galilean science of motion with a new

ontological and causal foundation. One is thus forced to ask why Mersenne was more strongly im-

pressed by the physical reasons speaking against the Galilean law than by those in its favor. Dear claims that the decisive factor was Mersenne's recognition that Descartes had managed to show that there was at least one "ingenious and plausible physical explanation of gravity that defied any neat mathematical analysis but un-

doubtedly compromised hopes of portraying Galileo's work as

genuinely demonstrative."" This explanation appears to me too weak. As I shall try to show in the following, Mersenne's eventual

scepticism was not due to the fact that Descartes had managed to

produce a "possible" explanation of gravity," but instead to the fact that he had formulated objections to Galileo's theory of acceleration which appeared to Mersenne valid irrespective of the valid-

ity of the Cartesian explanation. A comparison of Mersenne's

7 Dear, Mersenne, 215. 8 Ibid., 217. 9 Ibid., 215-216. 10 Ibid., 211. " Ibid., 218. 12 Ibid.




Novarum observationum tomus III (1647) with his Traite' des mouvemens (1634) conveys the impression that their author had moved from the original conviction that there existed many "possible" causal

explanations of gravity compatible with Galileo's law to the conclusion that there existed only one explanation-from innate gravity-which could, however, not be verified.

The following pages will be dedicated to a demonstration of how Mersenne began to doubt the validity of the odd-number law as soon as he became aware of the problematic nature of a fundamental principle of Galileo's kinematics. The principle in question is formulated both in the Dialogo and in the Discorsi and states that a falling body, dropped from a position of rest, has to pass through "infinite degrees of slowness." Mersenne, who was at first quite willing to accept the validity of this assumption and in fact even tried to furnish a demonstration of its truth, was later persuaded, in particular by Descartes, that it was incompatible with any strictly mechanistic explanation of gravity.'" For if one accepted the idea that bodies did not descend downward because of some intrinsic

property, but thanks to the action of an external force, one was forced to admit that they had to start moving with a finite speed and could not accelerate beyond a certain point. And this meant, in other words, that they could not accelerate according to the odd-number ratio.

My presentation of Mersenne's long-standing involvement with this issue will proceed chronologically and will include, under the various subheadings, detailed discussions of the authors with whom Mersenne stood in an epistolary exchange and of the crucial texts that influenced him. This procedure allows us to broaden the perspective beyond the figure of Mersenne and trace the

fortuna of the Galilean law of free fall in the middle of the seventeenth century.

1s Although Dear recognizes (ibid., 219) that the denial of the passage of the body through infinite degrees of speed was a crucial part of Fabri's, Baliani's and Descartes's respective theories of fall, he tends to exclude the possibility that Mersenne's doubts concerning the validity of the odd-number law could "arise from a rejection of the mathematical concept of a body passing successively through all degrees of speed from rest" (ibid., 211).




Mersenne's initial adherence to the Galilean position

We have said in the Introduction that Mersenne became only convinced of the odd-number law once he had carried out the exact measurements described in his Traite' of 1634. Yet it would be false to believe that his faith in experimental data was unlimited even then. For already in a passage in the Harmonie universelle to which Peter Dear has drawn attention, Mersenne compares Galileo's law of free fall with that of Gottfried Wendelin and points out that for small distances, the difference between the two laws is so small as to defy experimental verification.'" And he therefore concludes that "it doesn't suffice that three or four experiments in a row succeed to make a principle out of them."'5 Precisely because he was convinced that experiments alone could not "generate a science,"16 he sought in his Harmonie to develop mathematical arguments in favor of the Galilean law. And precisely in this context we may observe that he attaches great importance to the demonstration that bodies in free fall or descending along inclined

planes pass through infinite degrees of speed. The hypothesis that a heavy body that begins to descend from

an original position of rest has to pass through infinite degrees of

speed had been formulated in the first day of the Dialogo by Salviati who had tried to prove it by means of two propositions.'7 The first of them, which was presented as an undemonstrated truth, stated that "the degree of velocity acquired at a given point of the inclined plane is equal to the velocity of the body falling along the

perpendicular to its point of intersection with a parallel to the horizon through the given point of the inclined plane."'8 In other terms, the final speeds reached by two bodies descending along the inclined planes CA and DA (1) are identical to the two final

'4 Dear, Discipline, 132.

15 M. Mersenne, Harmonie universelle contenant la theorie et la pratique de la

musique, 2 vols., (Paris, 1636-1637), and facsimile reprint of author's annotated copy (Paris, 1963), 1: 126. The first volume of the Harmonie universelle, where this passage is to be found, was already printed in 1633, cf. R. Lenoble, Mersenne ou la naissance du micanisme (Paris, 1643), XXI-XXV.

16 Ibid., 112. 17 For a discussion of this proof, cf. also M. Clavelin, La philosophie naturelle de

Galilee. Essai sur les origines et la formation de la mechanique classique (Paris, 1996 [1968]), 287-290.

18 G. Galilei, Dialogue Concerning the Two Chief World Systems, tr. S. Drake, 2d ed. (Berkeley, 1967), 28 (=G.G., 7: 52).




7c

Fig. 1: G.G., 7: 51

speeds reached by two bodies in free fall down the verticals CB and DB, respectively.

The second proposition established that "in the ordinary course of nature, a

body with all external and accidental im-

pediments removed travels along an inclined plane with greater and greater slowness according as the inclination is less, until the slowness finally comes to be infinite, when the inclination ends by coincidence with the horizontal

plane.""9 This implies that one could choose, on the line CB, a

point so close to B "that if we were to draw a plane from it to the

point A, the ball would not pass over it even in a whole year." The truth of these two propositions necessarily entailed, in the eyes of Salviati,

that a falling body starting from rest passes through all the infinite grada- tions of slowness; and that consequently in order to acquire a determinate

degree of velocity it must first move a very great distance and take a very long time.20

The link between the premises and the conclusion is constituted

by a number of steps that Salviati fails to explain:

1) An infinity of inclined planes can be traced between the line CB and the point A.

2) Among these inclined planes, there exist no two planes at whose end the descending body would have reached the same degree of speed.

3) To each of the infinite degrees of speed reached at the end of the descent along one of the infinite inclined planes traced "above the horizon AD" corresponds a certain degree of speed through which the moving body must pass at some

point during its fall from C to B.

While going through his demonstration, Salviati declares that the

speed of free fall grows in a continuous manner, but he does not indicate the law according to which it does so. It is only in the second day of the Dialogo that he reveals that "the acceleration of

straight motion in heavy bodies proceeds according to the odd

'9 Ibid. 20 Ibid.




numbers beginning from one [...] which is the same as to say that the spaces passed over are to each other as the squares of the times."2' Salviati first explains this hypothesis in arithmetical terms and then goes on to invoke geometrical means which alone in his

eyes seems to be able to describe the continuity of acceleration (cf. fig. 2):

N~ A

a-------

Fig. 2: G.G., 7: 255

But since the acceleration is made continuously from moment to moment, and not discretely from one time to another, and the point A is assumed as the instant of minimum speed (that is the state of rest and the first instant of the subsequent time AD), it is obvious that before the degree of speed was acquired in the time AD, infinite others of lesser and lesser degree have been passed through. These were achieved during the infinite instants that there are in the time DA corresponding to the infinite points on the line DA [...]. Thus we may understand that whatever space is traversed by the moving body with a motion which begins from rest and continues uniformly accelerating, it has consumed and made use of infinite degrees of increasing speed corresponding to the infinite lines which, starting from the point A, are understood as drawn parallel to the line HD and to IE, KF, LG, and BC, the motion being continued as long as you please.22

Figure 2, which serves to illustrate the passage just quoted, shows the triangle of velocity ABC inscribed in a rectangle. Salviati em-

ploys this construction to show the so-called double distance rule which establishes that a body in uniform rectilinear motion passes in a given time twice the distance traversed in the same time by a

body in uniformly accelerated motion whose final speed is identical to the speed of the uniform motion of the first body. The demonstration produced by Salviati is based on a comparison between the surface of the triangle ABC, which represents the sum of the infinite degrees of speed of the accelerated motion, and that of the parallelogram AMBC, which represents "the total [or better: the mass] and aggregate of just as many degrees of speed but with each one of them equal to the maximum BC." But since the surface of the rectangle is twice that of the triangle, it seems "reasonable and probable" that a body "making use of the uniform velocities corresponding to the parallelogram [...] would pass with uniform motion during the same time through double the space which it passed with the accelerated motion.""23

Salviati presents his conclusion as "reasonable and probable" and not as "certain," because he perceives a difficulty in the identi-

21 Ibid., 222 (=G.G., 7: 249). 22 Ibid., 228-229 (=G.G., 7: 255). 23 Ibid., 229 (=G.G., 7: 256).




fication of the areas with the distances covered.24 It was in direct

response to this difficulty that Galileo later decided, in the Discorsi, to represent the space traversed in free fall by means of a straight line outside of the triangular area, thus leaving to the latter the task of representing the aggregate of the infinite degrees of veloc-

ity traversed by the falling body.25 A first mention of the notion of infinite degrees of speed occurs

in the Mechaniques de Galilee of 1634, in which Mersenne finds an occasion to slip in some personal "considerations regarding motions" which are quite evidently stimulated by his reading of the

Dialogo. "A falling body," he writes, "not only moves more slowly at the beginning of its motion, but it also passes through all the possible degrees of slowness."26

While this assumption occurs in the M&chaniques in a purely ca- sual manner, the Harmonie universelle of 1636 betrays already the author's growing awareness that it requires a demonstration. Mersenne here employs, albeit in inverted sequence, the two passages of the Dialogo summarized above. In the second proposition of its second book, which is dedicated to the mouvemens de toutes sortes de corps, Mersenne tries to prove that

if a heavy body, after traversing a given space [in a given time], did not further augment the speed it had acquired at the last point of this space and instead continued at the same speed, it would in an equal time cover a space twice as big as the first space. From this one may infer that the falling stone

passes through all possible degrees of speed.27

In order to demonstrate the theorem of the double distance, Mersenne uses figure 3, whose upper part reproduces the diagram printed in the Dialogo. And in his paraphrase of Galileo's words, Mersenne points out that since the speed of a body

-G

10

Fig. 3: Mersenne, Harmonie universelle, 1: 89

24 Concerning the problematic identification of aggregates of speed and spaces in the Dialogo, cf. M. Blay and E. Festa, "Mouvement, continu et composition des vitesses au XVIIe siecle," Archives Internationales d'Histoire des Sciences, 48 (1998), 65-118, esp. 75-76.

25 Cf. P. Damerow, G. Freudenthal, P. Mc Laughlin and J. Renn , Exploring the Limits of Preclassical Mechanics (New York, 1992), 229-230.

26 M. Mersenne, Les Mechaniques de Galilie, mathematicien et ingenieur du Duc de Florence avec plusieurs additions rares et nouvelles (Paris, 1634), 77-78; critical edition by B. Rochot (Paris, 1966), 71.

27 Mersenne, Harmonie universelle, 1: 89: "[...] si un grave estant tomb6 d'un

espace donne n'augmentoit plus la vitesse qu'il a acquise au dernier point de cet

espace, et qu'il continuait dans la mesme vitesse, il feroit un espace double du

premier en un temps egal: d'oi l'on infere que la pierre qui descend passe par tous les degrez possibles de tardivete."




grows continuously from one moment to the other, and not through pauses or jumps from one time to the other, it is certain that the degrees of speed [acquired] between the condition of rest A and the acquisition of the de-

gree HD in the time interval AD are infinite, according to the infinity of the instants of time AD or the points of the line AD.28

Without showing the slightest awareness of Galileo's own conceptual difficulties, Mersenne explicitly identifies the area of the tri-

angle ABC, which is constituted by the sum of the infinite degrees of speed of accelerated motion, with the space covered by the body in free fall.29 His reasoning thereby assumes the air of a certain

proof and not just of a probable argument, as in the case of Galileo's Dialogo. First, he compares the surfaces of the triangles AHD, AIE, AKF, etc., deducing that the spaces traversed by the

falling body are proportional to the squares of times elapsed. Then he compares the area of the triangle ABC with that of the parallelogram BCNO, concluding that the degree of speed BC reached at the end of the time AC would be sufficient for a body moved with a uniform speed to cover, in a time CO equal to AC, a space BCNO twice as large as ABC.30 After having repeated the same

argument in arithmetical terms-"to the advantage of those who do not know geometry"-Mersenne concludes by arguing

that the speed of falling bodies grows just in the same manner as time, for if after half a second the speed is 6, at the end of one second it will be 12, and at the end of one second and a half 18, and so on [...]. From this we can see that when we approach ever more closely the beginning of the fall, we can find such a great slowness of motion that a body that was to continue its fall with just that speed, would not be able to cover a space of one line in a thousand years: so that one can say that the body begins its fall with an almost infinite slowness; and that [the state of] rest can be considered com-

pletely infinite slowness, of which more will be said later.3'

28 Ibid.: "[...] croist continuellement de moment en moment, et non par pauses, ou sauts, de certain temps en certain temps, il est certain que les degrez de vitesse depuis le repos A iusques aI l'acquisition du degre HD dans le temps AD sont infinis, suivant I'infinite des instans du temps AD, ou des points de la

ligne AD." 29 A. Nardi has shown that not only Mersenne, but also Descartes and

Christiaan Huygens identified "without difficulty areas and distances." They did not experience Galileo's discomfort (more visible in the Discorsi than in the

Dialogo). Cf. A. Nardi, "La quadratura della velocitY: Galileo, Mersenne, la tradizione," Nuncius, 3, fasc. 2, (1988), 27-64, esp. 29, 61. For an analysis of the demonstration of the double distance rule in the Harmonie, cf. also id., "Spazi del moto in divina proporzione," Giornale critico della filosofia italiana, LXIII (1984), 334-376, esp. 343-344.

30 Mersenne, Harmonie universelle, 1: 90. 31 Ibid., 1: 91: "[...] que la vitesse des mobiles ne s'augmente qu'en la mesme




An occasion for returning to the same argument was offered by the demonstration of proposition VII whose aim was indeed "to

explain the motions of weights on inclined plains, to give the proportion of their speeds, and to determine if the falling weight passes through all possible degrees of speed.""2 Mersenne refers here once more to the Dialogo, mentioning that according to Galileo, a) two weights, beginning from an identical position, and one descending along an inclined plane and the other falling ver-

tically downward, will have identical "impetuosity" at points that are equally close to the center of the earth; and b) that the times of a perpendicular descent stand to the times of oblique descent in the same relation as the perpendicular to the oblique, and that (cf. fig. 3) the time of fall for AC stands to the time of fall along AB as the root of the space AC stands to the root of the space AB.

Aware of the fact that such conclusions were based on undemonstrable principles, Mersenne tried to subject them to an

empirical process of verification by measuring the times of fall

along planes of different lengths and angles of inclination. But the results he obtained were highly unsatisfactory, for it was "very difficult to perceive which of two balls lands first, when the first descends perpendicularly and the second along an inclined plane."33 Nonetheless, of one thing Mersenne seems to have remained convinced: acceleration required the passing through infinite degrees of speed. Upon the hypothesis that a body descending along the inclined plane BA (fig. 4) arrives at A with a speed sufficient to cross 30 feet in 100 years and that it reaches the same speed by descending vertically from B to C, Mersenne claimed that it followed that

faion des temps, car si apres une demie seconde la vistesse est comme 6, a la fin d'une seconde elle sera comme 12, & 'a la fin d'une seconde & demie elle sera comme 18, &c [...]. Par oii l'on void qu'en approchant toujours du commencement de la cheute, I'on peut rencontrer une si grande tardivet6 de mouvement, que le mobile ne feroit pas l'espace d'une ligne en mille ans, s'il continuoit 'a descendre de la mesme vitesse: de sorte que l'on peut dire qu'il commence sa cheute par une tardivete quasi infinie, et que le repos peut estre considere comme une tardivete entierement infinie, dont nous parlerons encore apres."

32 Ibid., 108-112, here 108: "expliquer les mouvements des poids sur les plans inclinez A l'horizon, avec la proportion de leurs vitesses; et determiner si le poids, qui tombe, passe par tous les degrez possibles de tardivet6."

ss Ibid., 112: "[...] tres-difficile d'appercevoir lequel tombe le premier des deux boulets, dont l'un tombe perpendiculairement, et I'autre sur le plan incline."




A4, c

Fig. 4: Mersenne, Harmonie universelle, 1: 110

it passes through all the degrees of slowness before acquiring a certain de-

gree of speed, given that one can incline the plane AB even less: and even if one takes the speed of the body at point E (which I suppose to be 3 /4 feet

away from B, for BE is equal to /4 of BA) the body would only traverse BE in 50 years and 15 feet in 100 years if it continued to move with this speed, which would also diminish the speed of the perpendicular fall BC in the same proportion.34

As becomes obvious in a passage of the Nouvelles pensees de Galime

(1639), which constitutes a free synthesis of Galileo's Discorsi intorno a due nuove scienze, Mersenne believed to have furnished, in the lines just quoted, a proof of the theory of infinite degrees of speed. For at the beginning of book IV of the Nouvelles pensees, which treats of the fall of bodies toward the center of the earth, the Minim explains that a naturally accelerated motion is one in which equal increases in speed are acquired in equal times;

And since there is an infinity of instants in every part of time, one will en- counter ever bigger degrees of slowness up to infinity as one approaches ever more closely the point of rest from which the moving body begins to descend, in such a manner that if a stone did not increase the speed it had

acquired in a certain time, it would not even traverse the length of one foot in a year, as I have shown in the seventh proposition of the second book on movements.35

In their critical edition of the Nouvelles pensees, Costabel and Lerner point out that Mersenne here echoes an argument used in the Discorsi by Sagredo when trying to demonstrate through numerical examples a truth at which human imagination usually

"4 Ibid., 110: "il passe par tous les degrez de tardivete, avant que d'avoir acquis un certain degre de vitesse, attendu que l'on peut ancor moins incliner le plan AB: et mesme si l'on prend la vitesse du mobile lorsqu'il est en E, que je suppose eloigne de B de trois pieds 3/4, car BE est le quart de BA, il ne fera BE qu'en 50 ans, et ne fera que 15 pieds en cent ans s'il continue dans cette mesme vitesse, laquelle fera aussi diminuer la vitesse de la cheute perpendiculaire BC en mesme proportion."

35 M. Mersenne, Les Nouvelles pensees de Galile, mathematicien et ingenieur du Duc de Florence...traduit d'italien en Franpais (Paris, 1639), critical ed. P. Costabel and M.-P. Lerner, 2 vols. (Paris, 1973), 1: 182: "Et comme il y a une infinite d'instans en chaque partie de temps, l'on peut en remontant vers le repos d'oii le mobile commence

' descendre, trouver des tardivetez tousjours plus grandes jusqu'9

l'infiny, de maniere que si une pierre n'augmentoit point la vitesse acquise dans un certain temps, elle ne descendroit pas la longueur d'un pied dans un an, comme j'ay demonstr6 dans la septiesme proposition du second Livre des Mouvemens." On Mersenne's paraphrasis of the Discorsi, cf. Boutroux, Le pire, 354-355; Lenoble, Mersenne, XXVI; J. Bernhardt, "Mersenne, commentateur de Galilee," Revue d'Histoire des Sciences, 28 (1975), 169-177; W.R. Shea, "Marin Mersenne: Galileo's <<traduttore-traditore>>," Annali dell'Istituto e Museo di Storia della Scienza di Firenze, 2 (1977), 55-70; Dear, Mersenne, 208-209.




balked, i.e. that the motion of fall begins with an extremely slow

speed.36 Evidently, the Minim felt that he had to reproduce also Salviati's demonstration which was based on the observation that the force of percussion of a falling body decreases with the dimi- nution of the height from which the body is dropped. But note that Mersenne summarizes Salviati's argument but cursorily and in fact criticizes Galileo for not having taken the pains of seeking the "true reason" behind natural acceleration. Not without some de-

gree of presumption, Mersenne refers those who would wish to fill the Galilean lacuna to the nineteenth proposition of book III of his own treatise.

The reader who checks this reference will find that this proposition is dedicated to the explanation of "various peculiarities of bodies which fall downward and of the speed of their fall.""' In the second corollary, Mersenne makes use of the double distance rule in order to show that if internal gravity adds to falling bodies a new

degree of speed in each moment of time it augments "their speed in the duplicate ratio of the times."38 But Mersenne's explanation ends on a note that will leave all those who hoped to find there "the true reason" of the motion of fall quite disappointed: "One can adapt this reasoning to the attraction of the earth or to the desire and propensity of bodies to join the rest of their kin.""39

The self-confidence displayed by Mersenne in the pages of his Nouvelles pensees was therefore not due to his belief to have found the one true cause of free fall, but instead to his conviction that there existed several causes, acting from within or without the fall-

ing body, that would produce an acceleration according to the odd-number law. But note that it was precisely with regard to this last point that Mersenne was to change his mind. We shall see in the course of the following pages how he gradually persuaded himself of the categorical difference in the effect produced, respectively, by internal or external forces. In fact, the conclusion he was to espouse towards the end of his life was that only a force acting

36 Mersenne, Les Nouvelles pensees, 2: 248-249.

7 Mersenne, Harmonie universelle, 1: 205. 38 Ibid., 208. For an analysis of Mersenne's reasoning, cf. the critical edition of

the Nouvelles pensees, 2: 250. s9 Mersenne, Harmonie universelle, 1: 208: "L'on peut accommoder ce raison-

nement A l'attraction de la terre, ou au desir, & ' la propension qu'ont les corps de se reunir avec leur tout." For an account of Mersenne's hypotheses concerning the cause of gravity, cf. Lenoble, Mersenne, 471-474.




from within and conferring upon the body "in each moment a new degree of motion" was capable of augmenting "its speed in the duplicate ratio of the times" and could bring about a passage of the body through infinite degrees of speed.

The impact of Descartes

The importance of Descartes' Principia philosophiae (1644) for Mersenne's change of mind on the subject of free fall has been discussed by Peter Dear, who rightly states that before the publication date of that work, Mersenne "had only a sketchy knowledge of Descartes's account of gravity."40 And yet, when we take a closer look at the correspondence between the two philosophers, we find that though Descartes gives away but few details of his own vortex

theory of gravity, he is quite clear and specific about the objections he levels against Galileo's theory of acceleration. Let us therefore return to the beginning of their discussion of this phenomenon.

In the fall of 1629, that is, even before Galileo had published the results of his research on the motion of free fall, Mersenne submitted to Descartes a question concerning the respective times of oscillation of pendula of various lengths. As he was not fully persuaded by the answer he received, he encouraged his interlocutor to explain the "foundations" of his view. Descartes, who had

just begun to compose Le Monde, thereupon decided to reveal to the Minim his hypothesis concerning the acceleration of falling bodies which he had elaborated some years earlier in the course of a stimulating exchange of views with Isaac Beeckman. In a letter of November 13, 1629, he explained to Mersenne that he was of the belief, first, "that a movement, once it is impressed on some

body, remains there perpetually, as long as it is not impeded by some other cause," and hence that "what has once begun to move in a vacuum will always move at an equal speed."41 Now, upon the

assumption that a body falling from A to C (fig. 5) received from

gravity "in single moments new forces towards its descent" and that at the same time it conserved the impetus it received in all preceding moments, one could conclude that it traversed the space AB in a period of time thrice as long as that required to traverse BC:

40 Dear, Mersenne, p. 217. 41 A.T., 1: 72; C.M., 2: 316.




But the proportion according to which this speed increases is demonstrated in the triangle ABCDE, where the first line denotes the force of the speed impressed in the first moment, the second line the force impressed in the second moment, the third line the third force, and so forth. Hence we get the triangle ACD which represents the increase of the speed of the downward motion of the weight from A to C, and ABE which represents the increase of the speed in the first half of the space which the weight traverses. We also get the trapezium BCDE which represents the increase of the speed

A's 'a 4

'I

LP

3 r C t1,,* Ia0

Fig. 5: A.T., 1: 72

in the second half [...], namely BC. And as the trapezium BCDE is three times larger than the triangle ABE-which is obvious -, it follows that the weight will fall three times more quickly from B to C than from A to B. In other words, if it needs three moments to pass from A to B, it will pass from B to C in a single moment.42

Though from the initial lines of this passage it might appear as if Descartes assumed that the falling body received from

gravity a new "vis celeritatis" in every successive moment of time, his figure represents a motion in which the velocity grows proportionally to space.43 For the cathetus

42 A. T., 1: 72-73; C.M., 2: 317: "Qua autem proportione augeatur ista celeritas, demonstratur in triangulo ABCDE: nempe prima linea denotat vim celeritatis impressam 10 momento, 2a linea vim impressam 2' momento, 3a vim 3' inditam, et sic consequenter. Unde fit triangulus ACD qui repraesentat augmentum celeritatis motus in descensu ponderis ab A usque ad C, & ABE qui repraesentat augmentum celeritatis in priori media parte spatii quod pondus percurrit, & trapetium BCDE quod repraesentat augmentum celeritatis in posteriori media parte [...] nempe BC. Et cum trapezium BCDE sit triplo maius triangula ABE, ut patet, inde sequitur pondus triplo celerius descensurum a B ad C quam ab A ad B: id est si tribus momentis descendit ab A ad B, unico momento descendet a B ad C."

4 Damerow, Freudenthal, Mc Laughlin and Renn have pointed out that Descartes tends to oscillate between a spatial and a temporal interpretation of the term "minimum." Against the view of those who, beginning with Koyre, attributed to Descartes the fixed idea that the speed of fall grew proportionally with space, these authors show that Descartes maintained, in the paper he drafted in 1618 in response to Beeckman's inquiry, that equal minima of force were added to the body in successive discrete minima of time and that therefore minima of motion increased in arithmetic progression. It was only in subsequent documents that we find the shift from a spatial interpretation of the minimum to a temporal one. According to these authors, "the implicit inconsistency of the spatial and temporal interpretations of minimum results necessarily from the application of the given conceptual framework to a new area of investigation." (Damerow et al., Exploring, 25-26). Descartes' analysis seems in fact to agree with the medieval theory of configurations of qualities and motions, where "applied to the extension [line AC in Fig. 5] [the] minimum is [...] indifferent to a temporal or spatial interpretation." (Ibid., 25).




AC designates the distance traversed by the falling body, while the areas ABE and BCDE indicate the increase of speed, respectively, in the first and second halves of this distance. By applying the medieval theory of proportions according to which "for equal spaces velocities are inversely proportional to times elapsed," Descartes arrives at the conclusion that the time required to cross the distance AB is three times longer than the time required for BC.44

When, in the summer of 1634, Descartes held for the first time a copy of Galileo's Dialogo in his hands, he did not notice that the theory of acceleration contained in it was incompatible with the one he had offered in his letter of 1629, for it defined speed as proportional to time rather than to space. In another letter to Mersenne, he wrote in fact that he had found in the Dialogo

some of my thoughts [...] which I think I have once communicated to you. The first of them is that the spaces traversed by heavy bodies when descending stand to one another as the squares of the times they take to descend, that is to say that if a ball takes three moments to descend from A to B, it will only take one moment to continue from B to C, etc. [cf. fig. 6]. This I said [was true] with many restrictions, for it is never entirely true in the way in which [Galileo] thinks it can be demonstrated.45

A

B -

C

Fig. 6: A.T., 1: 304

The nature of these "restrictions" had been

explained by Descartes a few years earlier. In a letter of June, 1631, he had asked Mer- senne to ignore the explanation of acceleration he had offered in his letter of Novem- ber, 1629, as it relied on two completely erro- neous assumptions. The first was that the motion of fall occurred in an absolutely void

space, while the second was that the velocity of the movement of fall was "at the first in-

stant the slowest that can be imagined and that it increases always uniformly thereafter.""46 Descartes had meanwhile come to believe

44 Ibid., 31. 45 Descartes to Mersenne, 14 August 1634, A.T., 1: 304-5; C.M., 4: 298: "[...]

quelques vnes des mes pens6es [...] queje pense vous auoir autrefois escrites. La

premiere est que les espaces par ou passent les cors pesans quand ilz descendent, sont les vns aus autres comme les quarres des tems qu'ils employent a descendre, c'est a dire que si vne bale employe trois momens a descendre depuis A iusques a B, elle n'en employera qu'vn a le continuer de B iusques a C, etc., ce que ie disois auec beaucoup de restrictions, car en effect il n'est iamais entierement vray comme il pense le demonstrer."

46 Descartes to Mersenne, 13 January 1631, C.M., 3: 23. In A.T., 1: 222, the same letter figures with the date of October, 1631.




that "the true proportion defining the increase of the speed of a

weight descending through air" could only be given once the nature of gravity had been determined.

But then, it was precisely as he initiated his attempts to develop his own physical explanation of the motion of fall that Descartes understood how difficult it was to arrive at a universally valid law of acceleration. In a letter to Mersenne, written toward the end of 1632, Descartes stated that according to the principles of his own natural philosophy, the respective speeds of two falling globes of lead, weighing one pound the first and hundred pounds the second, should stand in a different ratio than, say, the speeds of two wooden balls of one and hundred pounds, or of two leaden balls of two and two hundred pounds, respectively.47 As the editors of the Mersenne Correspondence point out, this was the result of Descartes' notion that terrestrial bodies were predominantly made

up of particles of the third element with a varying quantity of subtle matter interspersed in their pores. Since, according to the

theory developed in Le Monde, terrestrial bodies were pressed towards the earth by the vortices of subtle matter, and since these vortices exercised their push only on the particles of the third

type, the physical make-up of a body became obviously a decisive factor in determining its acceleration.48

But as Mersenne was in 1632 not yet acquainted with the physical theory of acceleration that Descartes was at the time elaborat-

ing for his Le Monde, he may have thought that the letter of his interlocutor referred merely to the different degrees of resistance offered by the air to falling bodies of diverse shape and weight. Such a resistance was, after all, also admitted by Galileo, though the latter, in contradistinction to Descartes, regarded it as a mere accident one could abstract from.

At the time when Mersenne wrote his pages on the motion of

falling bodies analyzed in our preceding section, he could thus have grasped only partially the reasons behind Descartes' declara- tion that Galileo's odd-number law was unacceptable.49 Some ad-

47 A.T., 1: 261; C.M., 3: 344-345. 48 C.M., 3: 347. 49 In a letter sent to Mersenne probably in 1635, Descartes alluded to his

numerous reasons for believing that Galileo's law of acceleration was false, but he limited his discussion to the resistance of air: "Car cette proportion d'augmentation selon les nombres impairs 1, 3, 5, 7 etc., qui est dans Galilee, et

queje croy vous avoir escritte autrefois, ne peut estre vraye , commeje pense vous




ditional years were necessary for him to understand fully what Descartes had tried to express when he wrote, in his letter of June 1631, that the velocity of fall was neither the smallest conceivable at its very beginning nor that it grew in a continuous manner. The first extant letter in which Descartes offers Mersenne an explanation of his own theory of weight carries in fact only the date of

June 19, 1639:

- -.- -.. C ?

C

Ao

D

,

----,. _

Fig. 7: A.T., 2: 565

Regarding what you wrote to me about the problem of weight, the stone C is pushed in a circle by the subtle matter and hence towards the center of the earth; but the first [circular] motion cannot be perceived, because it is shared by the entire earth and the surrounding air, so that we are left with the second [centripetal motion] as a cause of weight. And this stone moves more quickly toward the end of its fall than at the beginning, though it is

pushed less forcefully by the subtle matter: for it retains the impetuosity of its precedent motion, and whatever the action of the subtle matter adds to it leads to its increase.50

But if heavy bodies fall downward as an effect of the push of subtle matter, it follows that their motion should obey two general laws

governing collision. The first establishes that "if one body moves another, it must lose as much of its movement as it gives to the other.""51 According to the second, "the larger the bodies are, the

avoir aussi mande alors, qu'en supposant deux ou trois choses qui sont fausses. Dont l'une est que le mouvement croisse par degrez depuis le plus lent ainsi que juge Galilee, et I'autre que la resistance de l'air n'empesche point. Et cette derniere cause peut faire que les cors qui descendent, estant parvenus a certain

degre de vitesse ne l'augmentent plus et ceux qui sont de matiere fort legere, parviennent plus tost a ce degre de vitesse que les autres." (C.M., 5: 581). Inci-

dentally, this remarkable passage flies in the face of Descartes' usual denial of Beeckman's idea that due to the resistance of the air the motion of the falling body reaches a certain punctum aequalitatis in cadendo. On Descartes' disagreement with Beeckman, cf. Damerow et al., Exploring, 41 and Lenoble, Mersenne, 468, where the relevant passages are cited. Incidentally, Lenoble offers a detailed account of Mersenne's changing views on the existence of a punctum aequalitatis in cadendo, cf. ibid., 468-470.

5o A. T., 2: 565; C.M., 8: 454-455: "Touchant ce que vous m'escrivez de la pesanteur, la pierre C est poussee en rond par la matiere subtile, & avec cela vers le centre de la terre; mais le premier est insensible, a cause qu'il est commun a toute la terre, & a l'air qui l'environne, si bien qu'il ne reste que le second qui fait la pesanteur. Et cete pierre se meut plus vite vers la fin de sa descente qu'au commencement, bien qu'elle soit poussee moins fort par la matiere subtile: car elle retient l'impetuosite de son mouvement precedent, & ce que l'action de cete matiere subtile y adiouste l'augmente." For a clear account of Descartes' theory of gravity, cf. E.J. Aiton, The Vortex Theory of Planetary Motions (London, 1972), 55; R.J. Overmann, Theories of Gravity in the Seventeenth Century (Ph.D. thesis, Indiana

University, 1974), 80-104. 51 Descartes to Mersenne, 28 October, 1640 (A.T., 3: 211; C.M., 10: 173): "si




slower they should move when pushed by the same force."52 When viewed in the light of these two premises, Descartes' ear-

lier criticism of the Galilean theory of acceleration finally had to

acquire a new meaning for Mersenne. For assuming that heavy bodies were pushed downward by subtle matter (or by some other force acting by contact), it was necessary to conclude, first of all, that they could not pass through infinite degrees of speed but had to begin their motion with a determined degree of speed. The

principle of the conservation of the quantity of motion implied that the loss of motion on the part of the moving body was being compensated instantaneously by an equal acquisition of motion on the part of the body moved. From the same principle, one could have derived also another consequence, namely that bodies of different sizes accelerated differently, as Descartes pointed out to Mersenne in November 13, 1639:

I believe that in the void, if one such could exist, the smallest force could move the largest bodies just like the smallest, but not with the same speed. For the same force would make a body with twice the size move half as fast as that of the simple size.53

un cors en meut un autre, il doit perdre autant de son mouvement qu'il lui en donne." In a letter of April 30, 1639, sent to De Beaune by way of Mersenne, Descartes derives the same law from the principle of the constancy of the quantity of mouvement in created matter: "Premierement, ie tiens qu'il y a une certaine Quantite de mouvement en toute la Matiere crete, qui n'augmente, ny ne diminuE iamais; et ainsi, que, lors qu'un corps en fait mouvoir un autre, il perd autant de son mouvement qu'il luy en donne." (A.T., 1: 543; C.M., 8: 421). This law is presented as the third law of nature in the Principia (A.T., 8/1: 65) and as the second law in Le Monde (A. T., 11: 41). On these laws of motion, cf. D. Garber, Descartes' Metaphysical Physics (Chicago, 1992), 197-231; D. Des Chene, Physiologia: Natural Philosophy in Late Aristotelian and Cartesian Thought (Ithaca, 1995), 272-312; M. Blay, "Les regles cartisiennes de la science du mouvement dans Le Monde ou traits de la lumiere," Revue d'Histoire des Sciences 51 (1985), 319- 346.

52 Descartes to Mersenne, 25 December, 1639 (A. T., 2: 627; C.M.,8: 696): "plus les corps sont grands, plus ilz doibuent aller lentement, lorsqu'ilz sont poussez par une mesme force." This law, too, had been formulated in Descartes' letter to De Beaune of April, 1639: "Et pour ce que, si deux cors inegaux regoivent autant de mouvement I'un que l'autre, cette pareille quantit6 de mouvement ne donne pas tant de vitesse au plus grand qu'au plus petit, on peut dire, en ce sens, que plus un cors contient de matiere, plus il a d'inertie naturelle; a quoy on peut adjouter qu'un cors qui est grand, peut mieux transferer son mouvement aux autres cors, qu'un petit, et qu'il peut moins estre mfi par eux. De faeon qu'il y a une sorte d'inertie, qui depend de la quantite de la matiere, et une autre qui depend de l'estenduE de ses superficies." (A.T., 2: 543-544; C.M., 8: 421).

3 A.T., 2: 623; C.M., 8: 611: "Ie croy bien que dans le Vuide, s'il estoit possible, la moindre force pourroit mouvoir les plus grands cors, aussi bien que les plus petits, mais non de mesme vitesse. Car la mesme force feroit mouvoir une




It is, however, not "the same force" to exercize its power on bodies of different sizes, as we gather reading Le Monde and the

Principia. In both treatises it is stated clearly that the volume of subtle matter pushing a heavy body towards the center of the earth is always identical to the volume of the body itself.54 And yet, as we have pointed out earlier, Descartes had different grounds for be-

lieving that the acceleration of free fall could not be the same for all bodies. As we recall, he held that, of two bodies of equal size but different composition, the one with a lesser quantity of subtle matter in its pores had to fall faster than the other.55

When Descartes affirmed, in his letter of June, 1631, that the

velocity of fall did not always increase equally, he probably did not

only wish to express that bodies of different mass varied in their

respective acceleration, but also that none of these bodies had a uniform acceleration. In a letter to Mersenne of March 11, 1640, he admitted in fact his "inability to determine" either "the speed with which any heavy body falls at the beginning, since this is a

merely factual question which depends entirely on the speed of the subtle matter,"56 or the proportion according to which this initial speed was made to increase after every single collision with the subtle matter:

The subtle matter pushes in the first instant the descending body and gives it one degree of speed; then in the second instant it pushes it a little less and gives it almost one other degree of speed, and so on in the other [instants]. This happens more or less in a duplicate ratio at the beginning of the fall. But this proportion gets entirely lost after the bodies have fallen several fathoms and the speed does not increase any more, or almost not at all.57

pierre double en grosseur, de la moitie moins viste que la simple." The idea that "a minimal force can move also the largest bodies" was abandonned by Descartes in his Principia, where he states that a body, irrespective of its velocity, is unable to move a body larger than itself. On Descartes' changing view on this matter, cf. Des Chene, Physiologia, 298-299.

54 Cf. A.T., 8/1: 213-214; A.T., 11: 76-77.

55 This view is explained clearly in the Principia, A.T., 8/1: 214. 56 A.T., 3: 36; C.M., 9: 190. 57 A. T., 3: 37-38; C.M., 9: 191-2: "La matiere subtile pousse au premier moment

le cors qui descend, & lui donne un degr6 de vitesse; puis au second moment elle

pousse un peu moins, & luy donne encore presque un degre de vitesse, & ainsi des autres; ce qui fait fere rationem duplicatam, au commencement que les cors descendent. Mais cette proportion se perd entierement, lorsqu'ils ont descendu

plusieurs toises, & la vitesse ne s'augmente plus, ou presque plus." What is ex-

plained in this letter to Mersenne is discussed in greater detail in a manuscript written by Descartes in 1635 and published in the Excerpta Anatomica in




The existence of a punctum aequalitatis in cadendo--a point at which the falling body ceases to accelerate further-which in a letter of 1635 Descartes had presented as a possible effect of the impeding action exercized by the medium,58 is now justified as the conse-

quence of the following dynamic principle: A decrease in the difference between the respective speeds of the pushing and the

pushed bodies implies a concomitant decrease in the ability of the first to act on the second, which in turn implies that the speed of the pushing body represents the limit of the speed of the body being pushed. As Descartes explains in a later letter, written in

June 11, 1640, this means that the acceleration imparted by the subtle matter on the falling bodies by each successive impact must diminish progressively until it disappears completely at the point where the falling body has reached the very speed of the subtle matter.59

We have seen that Descartes was not able to convert his mechanical explanation of free fall into a law of acceleration. This

inability had to, first, with the impossibility of measuring the

proper speed of the subtle matter and, second, with the difficulty of determining the effect that the latter was capable of producing with each successive impact. To these reasons must be added a third which Descartes never explicitly indicated but which can be inferred from a letter he sent to Mersenne in January, 1640. Wish-

ing to convince his correspondent of the fact that falling bodies could not pass through an infinity of degrees of speed, Descartes wrote:

I have just looked once more through my notes on Galileo, where I did not really state that falling bodies do not pass through all degrees of speed, but I said that this cannot be determined without knowing what weight actually is, which comes to the same. As for your example of the inclined plane, it certainly proves that all speed is infinitely divisible, but not that when a body

A.T., 11: 621-634. This manuscript is discussed in Damerow et al., Exploring, 41- 42.

5s Cf. the quotation at fn. 49.

5 A. T., 3: 79; C.M., 9: 398-9. It is interesting to see that a similar kind of reasoning is found in a page of Beeckman'sJournal, where its author tries to determine what would happen, in the void, if an atom hit repeatedly two globes, one of which very heavy, the other very light. Beeckman reaches the following conclusion: "Primus impetus plus promovebit levissimum quam graviorem; sic etiam secundus etc., donec ad atomorum, eadem semper celeritate motarum ac de novo perpetuo impactarum, celeritatem pervenerit. Gravior tandem quidem ad eam celeritatem, sed serius perveniet; quamdiu enim non tam celeriter movetur quam atomus, semper hujus impetus aliquid celeritati adijciet." (I. Beeckman, Journal, 3: 131, September, 1629).




begins to descend, it passes through all these divisions. And when one hits a ball with a hammer, I do not think that you assume that this ball at the

beginning of its movement moves more slowly than the hammer, nor that those bodies that are pushed by others fail to move from the very first moment onward with a speed proportional to that of the bodies moving them. Now, according to me, weight is nothing else than terrestrial bodies being actually pushed toward the center of the earth by the subtle matter, from which you can easily gather the consequence. And yet, one must not therefore assume that the bodies move at the beginning as quickly as this subtle matter itself; for the latter pushes them only obliquely, and they are very much impeded by the air, in particular the lighter ones."

Descartes here explains the extreme slowness with which bodies accelerate at the beginning of their fall with the fact that they are

pushied only obliquely by the subtle matter-but then, we know that Descartes never managed to formulate a theory of oblique collisions. This means, however, that even under the assumption that he had found the two magnitudes he had been looking for- to wit: the speed of subtle matter and the diminishing quantity of

velocity it conveyed to the falling body in each collision-he would still not have been able to determine the ratio of acceleration.6'

6 Descartes to Mersenne, 29 January, 1640, A.T., 3: 9-10; C.M., 9: 89: "Je viens de revoir mes Notes sur Galilee, oiije n'ay veritablement pas dit que les cors qui descendent ne passent pas par tous les degrez de tardivete; mais j'ay dit que cela ne se peut determiner sans scavoir ce que c'est que la pesanteur, ce qui signifie la mesme. Pour votre instance du plan incline, elle prouve bien que toute vitesse est divisible t l'infiny, mais non pas que lorsq'un cors commence a descendre, il

passe par toutes ces divisions. Et quand on frappe une boule avec un mail, je ne

croy pas que vous pensiez que cette boule, au commencement qu'elle se meut, aille moins vite que le mail; ny enfin que tous les corps qui sont poussez par d'autres, manquent a se mouvoir, des le premier moment, d'une vitesse

proportionee 'a celle des cors qui les meuvent. Or est-il que, selon moy, la

pesanteur n'est autre chose, sinon que les cors terrestres sont poussez reellement vers le centre de la Terre par la matiere subtile, d'oui vous voyez aisement la conclusion. Mais il ne faut pas penser, pour cela, que ces cors se meuvent au commencement si viste que cette matiere subtile; car elle ne les pousse qu'oblique- ment, et ils sont beaucoup empechez par l'air, principalement les plus legers."

61 On the lack of a theory of oblique collision in Descartes' philosophy, cf. Garber, Descartes, 357-358; and Damerow et al., Exploring, 120-123, whose authors draw attention to what seems to be the only case in which Descartes deals in a

quantitative way with a case of oblique impact. This happens in a letter to Mersenne of April, 1643, where Descartes claims that a moving ball colliding obliquely with a smaller ball at rest should make the latter move faster than it itself moves (A. T., 3: 652; C.M. 12: 159). At first sight this statement could appear to be in contradiction with the passage of the letter of January, 1640 quoted above, where Descartes claims that subtle matter, when colliding obliquely with

heavy bodies, imparts to them a speed that is inferior to the speed it would impart to them in a straight collision. But in fact the two cases are different. As we have

already mentioned, for Descartes the volume of subtle matter pushing the body




The reference to the infinite divisibility of the speed of fall, which is contained in the letter just cited, appears particularly relevant in the present context. For it shows that the argument from inclined planes put forth in Galileo's Dialogo and further developed in Mersenne's Harmonie universelle seemed to Descartes only inconclusive from a physical point of view, but not false from a mathematical point of view. For while it demonstrated the existence of infinite degrees of speed mathematically, it did not prove that a body in descent actually did pass through each and every of these infinite degrees. On this issue, Descartes' view was, interest-

ingly enough, diametrically opposed to Galileo's. For the latter

explained, in the Dialogo, that he did not believe "that it was impossible for nature or for God to confer immediately" a determined speed to some heavy body, but "that defacto nature does not do so-that the doing of this would be something outside the course of nature, and therefore miraculous."'' Descartes, to the

contrary, argued in a letter to Mersenne of 1638: "What Galileo says, that descending bodies pass through all degrees of speed, I don't think that it happens this way ordinarily, but that it is not impossible that it happens sometimes.""63

In other words, both authors, while taking opposite stances on the ordinary course of events, did not bar the possibility that the

contrary case could take place. Descartes, it seems, did not wish to exclude the possibility of an infinite divisibility of speed for the same reason for which he would also insist on the infinite divisibility of matter: in both cases, the power of God had to remain without limits. This attitude explains also why he was not ready to accept a logico-mathematical argument employed by "Monsieur F." in a confutation of the Galilean hypothesis regarding the actual passage of a mobile through infinite degrees of speed:

There is something unsatisfactory about the argument used by Monsieur F. to refute Galileo, when he says that "speed is acquired, either in a first instant, or in any determined time," for neither one nor the other is true. In the terminology of the Schools, one can say that "it is acquired in time, where time is taken inadequately.""64

toward the earth is perfectly equal to the volume of the body itself and not smaller as in the case discussed in the letter of April, 1643.

62 Galilei, Dialogue, 21 (=G.G., 7: 45). 63 Descartes to Mersenne, 11 October 1638, A.T., 2: 399; C.M., 8: 114: "Ce que

dit Galilee, que les cors qui descendent passent par tous les degrez de vitesse, je ne croy point qu'il arrive ainsi ordinairement, mais bien qu'il n'est pas impossible qu'il arrive quelquesfois."

64 Ibid.: "Et il y a du m&conte en l'argument dont se sert M. F. pour le [= ce




The "Monsieur F." mentioned in these lines is none other than Pierre Fermat, who in a treatise that was only published in 1922 by Cornelis De Waard had attempted to demonstrate that the Galilean hypothesis of infinite degrees of speed contradicted the

postulate that "there can be no motion without some speed of a moved body" ("nullum motum fieri absque celeritate aliqua cor-

poris moti").65 Fermat reasoned on the assumption that a falling body could only acquire speed either "in primo instanti" or "in determinato tempore." But since the second possibility, which was the one embraced by Galileo, stood in contradiction to the above- mentioned postulate, one could not but choose the alternative view that the heavy body "when it begins to move possesses speed" ("cum incipit moveri habet celeritatem").66 It could therefore not

pass through an infinite number of degrees of slowness before

acquiring that speed. But Descartes believed that there existed a third possibility,

namely that the falling body, beginning from a position of rest, acquired speed "in tempore inadaequate sumpto." In one of his

subsequent letters to Mersenne, Descartes explained that "By quantitas inadaequate sumpta, I understand a quantity that though possessing in fact all of its three dimensions should not always in the case at hand be considered as if it had them."''67 Though this definition is everything but clear, the only plausible interpretation seems to be that for Descartes, it was in principle possible that in order to acquire some finite speed, a body employed a quantity of time that was smaller than any "determined time," but larger than an unextended instant. And yet, as he repeatedly declared in his letters to Mersenne between 1640 and 1642, he was convinced that, as a matter of ordinary fact, the body acquired a determined

degree of speed already in the first instant of its fall. And the ar-

guments he used to persuade his interlocutor of this point invari-

ably appealed to the "shock" between colliding bodies. In a letter

que dit Galilee] refuter, en ce qu'il dit que acquiritur celeritas, vel in primo instanti, vel in tempore aliquo determinato car ny l'un, ny I'autre n'est vray, et en termes d'Escole on peut dire que acquiritur in tempore inadaequate sumpto."

65" crit anonyme inedit sur la chute des graves, in Oeuvres de Fermat, Supplement aux tomes I-IV, published by C. De Waard (Paris, 1922), 36.

66 Ibid., 37. 67 Descartes to Mersenne, 15 November 1638, A.T., 2: 445; C.M., 8: 209: "Per

quantitatem inadaequate sumptam, j'entens une quantite qui, bien qu'elle ait en effet toutes ses trois dimensions, ne se considere pas toutesfois au cas propose comme les ayant."




A1 AB

Fig. 8: A.T., 3: 593

of 1642, Descartes asked Mersenne to

imagine a large cannon ball (A) striking against a little suspended ball (B) (cf. fig. 8).

Now, if one admits that the body A suc- ceeds in setting the body B into motion, then one has to admit "by the same to-

ken" also that B moves from the first instant with the same velocity of A and that it therefore does not pass through infinite degrees of speed. For if in the very first moment subsequent to the collision, B moved very slowly, A-which follows B in close contact- would be forced to slow down. But this decrease in speed would have to be definitive, for

there is no reason that would make it resume afterwards its former speed, since the gunpowder, which had initially propelled it, is no longer active; and when a body has been for one moment without moving or moving very slowly, this amounts to the same as if it had not been moving for a longer period."

The fact that Descartes suggested the use of a cannon ball to simu- late the action exercized by the subtle matter agrees perfectly with the views he had expressed in the letter of March 11, 1640, which we have already had occasion to cite:

I do not distinguish between violent and natural motions; for what difference does it make if a stone is pushed by a man or by subtle matter? And thus, once one admits that violent motions do not pass through all the degrees of speed, it seems to me that one has to admit the same for natural motions.69

In his study on seventeenth-century mechanics, Rene Dugas claims that Galileo's identification of a body's tendency to fall with its resistance to rise had been sufficient to destroy the Aristotelian distinction between the intrinsic principle of the motion of fall and the extrinsic principle of projectile motion.7v The case of

' Descartes to Mersenne, mid-November, 1642, A.T., 3: 593; C.M., 9: 349-350: "il n'y aura point de raison qui luy face par apres reprendre sa premiere vitesse, a cause que la poudre

' canon, qui l'avoit pousse, n'agist plus; et quand un corps

a este un moment sans se mouvoir, ou en se mouvant fort lentement, c'est autant que s'il y avoit est' plus long temps."

69 Descartes to Mersenne, 11 March 1640, A.T., 3: 39; C.M., 9: 193: "Je ne mets aucune difference entre les mouvemens violens et les naturels; car qu'importe, si une pierre est poussee par un homme, ou bien par la matiere subtile? Et ainsi, avofiant que les violens ne passent pas par tous les degrez de tardivet6, il faut, ce me semble, avoiler le mesme des naturels."

70 R. Dugas, La mecanique au XVII sikcle (Paris, 1954), 67.




C

B D

A

Fig. 9: A.T., 3: 593

Descartes shows, however, that the converse is not necessarily true: though taking re- course to external principles to account for both free fall and projectile motion, Des- cartes was unwilling to accept the Galilean

equation between downward acceleration and upward deceleration. In a letter to Mer- senne of April, 1643, he expressed his conviction that an arrow, for example, used up less time to rise from A to C than to descend from C to E (fig. 9).7 The reason for this

asymmetry lay for him in the fact that the deceleration of the arrow between A and C was uniform while its renewed acceleration from C to E was not, as the speed of fall in-

creased more rapidly from C to D than from D to E. In other words, while in Galileo, there exists only a downward

directionality which works symmetrically on rise and fall, in Descartes space itself is matter in motion and therefore influences rise and fall asymmetrically.

The letters analyzed in this section, which cover a time span of more than ten years, furnish a good illustration of Descartes' scepticism concerning the validity of the principles of Galileo's new science of motion. Mersenne, who had become a convinced adept of this science as soon as he had first laid eyes on the Dialogo, began to realize only slowly the implications of the criticism voiced

by his correspondent. Admittedly, Descartes had at first only un- derlined the inadequacy of the Galilean theory of acceleration for a world in which heavy bodies fell not through a void, but through a medium. But he subsequently initiated his attempts to persuade Mersenne that even in an imaginary void, there could exist no force such as to accelerate all bodies according to the ratio indicated by Galileo. Because of the principle of the conservation of the quantity of movement, one was forced, so he claimed, to assume that all heavy bodies received a determined degree of speed at the very moment at which they collided with the subtle matter and that this speed increased with each successive collision, the limit of acceleration being given by the very speed of the pushing

" Descartes to Mersenne, 26 April 1643, A.T., 3: 657; C.M., 12: 164-165.




medium. The motion of fall was therefore not uniform. Instead, one had to assume that the higher the speed of the falling body, the less the effect exercized on it by the little pushes of the subtle matter.

In sum, to postulate in the face of Galileo that gravity was not an intrinsic property of bodies but instead the effect of forces

working from without meant, for Descartes, to reject the hypothesis of the continuity of acceleration, the odd-number law as well as the symmetry of the trajectory of projectiles.

Mersenne's first doubts

While Descartes believed that the phenomenon of free fall was

governed by too many variables to be translatable into mathematical laws, some of his contemporaries tried instead to substitute Galileo's law with alternative mathematical formulae. In October, 1643, Mersenne wrote a letter to Theodore Deschamps which is

unfortunately no longer extant but in which he mentioned two of these alternative laws.72 The first of them, which had been elaborated by Honore Fabri, assumed that the spaces traversed by the falling body in successive equal times grew according to the series of natural numbers (i.e. 1, 2, 3, 4, ...). The second of them, which had been thought up by another Jesuit, Pierre Le Cazre, proposed that the spaces grew according to the series of ever dou-

bling numbers (i.e. 1, 2, 4, 8, ...). In his answer, dated November

1, 1643, Deschamps offered, however, an interesting a priori proof in favor of the superiority of Galileo's odd-number law (i.e. 1, 3, 5, 7, ...):

Now, in reply to your letter, it seems to me that [neither] the natural progression of numbers, nor the geometrical double manage to measure acceleration. For apart from the fact that experience contradicts them, assume the following case: that in one moment of time the body descends by a certain measure, and in two, three and four equal moments of time, the number of measures follows one or the other progression. Now, if one chooses for the time of the first space another time than the one formerly chosen, for example its double or triple, the spaces traversed will follow no longer either the one or the other progression, as they do, by contrast, in the progression of odd numbers. Which is easily demonstrated.'"

72 On Deschamps' relation with Mersenne, cf. Lenoble, Mersenne, 423-425. 73 C.M., 12: 351: "Maintenant pour respondre a vostre lettre, il me semble que

la progression naturelle des nombres, ni la geometrique double, ne peuvent




Deschamps was correct in pointing out that Galileo's law of acceleration was the only one to possess a property that we would nowa-

days call scalar invariance, which means that it remains valid for whatever basic unit is chosen to measure the times of descent.

Deschamps' reasoning can be exemplified as follows: imagine that a heavy body in free fall passes in successive and equal intervals of time t distances equal to

either a) is, 2s, 3s, 4s, 5s, 6s (Fabri)

or b) is, 2s, 4s, 8s, 16s, 32s (Cazre)

or c) is, 3s, 5s, 7s, 9s, Ils (Galileo)

If one chooses to assign a duration of 2t to the successive instants of time, the distances traversed will be as follows:

a') 3s, 7s, 1ls (Fabri) b') 3s, 12s, 48s (Cazre) c') 4s, 12s, 20s (Galileo)

If one compares the series a', b', c' with the series a, b, c, one will at once be able to observe that the ratio between the distances crossed in successive and equal times, beginning from rest, will remain invariable only according to the law given by Galileo. The growth of the spaces traversed in equal times follows in a' a progression that is different from that of the natural numbers (1s:2s:3s ? 3s:7s:11s), and in b' a progression that is different from that of the ever doubling numbers (1s:2s:4s ? 3s:12s:48s). In the case c', however, the ratio of odd numbers is preserved (1s:3s:5s = 4s:12s:20s). In other words, Cazre's and Fabri's respective theories of acceleration required certain determinate units of time in order to function.

mesurer l'acceleration. Car outre l'experience contraire, mettant le cas qu'en un

temps le grave descendit une certaine mesure, et en deux, en trois et en quatre semblables temps, le nombre des mesures port6es par l'une ou l'autre progression, si on vient a prendre pour le temps du premier espace, un autre temps divers d'iceluy, comme, par example, double ou triple, les espaces parcourus ne

garderont plus ni l'une ni l'autre progression, comme ils font en la progression des nombres impairs. Ce qui est facile a demonstrer."




Mersenne's reply to Deschamps is unfortunately once more lost to us. As we shall see further below, when Le Tenneur and

Huygens later defended the Galilean theory of acceleration with

arguments very similar to Dechamps', Mersenne remained unconvinced. His counter-argument would in fact be that the va-

lidity of the odd-number law could not be demonstrated on the basis of mathematics alone, but needed to be derivable from a

physical explanation. Yet, a passage of the ballistic section of the

Cogitata physico-mathematica of 1644 seems to indicate that, at least

initially, Mersenne was impressed by Deschamps' arguments. There, the author explains that the Galilean law was not only in

greater conformity with experience than either Fabri's or Cazre's laws, but that it was also confirmed by rational arguments. For this reason, Mersenne continued to support the odd number law-at least for the time being and while waiting for Descartes to work out his own theory of acceleration:

Since therefore that progression of ours by odd numbers [...] has seemed always to correspond to our experience, and is confirmed by its moments of reasons, we will retain it until another progression will be demonstrated by the illustrious sir, who though he does not believe that heavy bodies pass through all degrees of speed from the point of rest [...], yet says that this progression is almost true.74

It is particularly noteworthy that in the eyes of Mersenne, it was

precisely the unwillingness to accept the hypothesis that a falling body passed through infinite degrees of speed to have constituted Descartes' main reason for dissenting from Galileo. Mersenne

obviously believed that the truth of this hypothesis also implied the truth of the odd-number law. At the same time, however, he

74 M. Mersenne, Cogitata physico-mathematica (Paris, 1644), Phaenomena ballistica, 52: "Cum igitur illa nostra per numeros impares progressio [...] semper experien- tiae nobis respondere visa sit, suisque rationum momentis confirmetur, eam retinebimus, donec alia demonstrata sit ab Illustri viro, qui licet gravia credat non transire per omnes tarditatis gradus a puncto quietis [...], fatetur tamen hanc progressionem esse proxime veram." I interpret the phrase "suisque rationum momentis" differently than Dear, who takes it to refer "to the account of uniform acceleration in free fall first communicated to him by Beeckman, involving the successive compounding of acquired downward impulses by the mobile" (Dear, Mersenne, 213, fn. 51). Instead, I tend to think that "momentum" is here employed in a figurative sense. Both in the Dialogue and in the correspondence, Galileo uses the expression "momenti di ragioni" to signify the "importance of reasons" (cf. P. Galluzzi, Momento. Studi galileiani (Rome, 1979), 416) or "the validity of the arguments" (Galilei, Dialogue, 369, Drake's translation). This reading finds support in the Novarum observationum, where Mersenne judges Le Tenneur's mathematical arguments in favor of the Galilean law insufficient in spite of what he calls the "rationum praestantia" (cf. fn. 146, below).




seemed to maintain that this law could nevertheless remain valid, if only "approximately," also in a world in which bodies began to fall with a determinate speed:

When I said that the falling or rising chord [of a pendulum] passes through all degrees of speed, I do not wish that you take this assertion as something demonstrated, but rather as something probable and recognized by Galileo, given that eminent philosophers and mathematicians deny it, and though bodies can possess a certain degree of speed with which they begin to move about the center, as would probably happen if the descent of stones and other heavy bodies was produced by the attraction of the earth or by expul- sion in such a way that, just as water expels lighter bodies, so heavier things expel air or some other matter which is even more subtle than air, which through its turn-about and perpetual revolution shakes off or drives away stones and things of this kind. All of which does, however, not prevent heavy bodies from increasing their velocity in the duplicate ratio of the times, nor will the sense detect any evidence to the contrary in observations.75

With these lines, Mersenne implicitly admits that the proof he had offered in the Harmonie universelle of the hypothesis that a body passes through infinite degrees of speed was inconclusive. More

importantly, he seems to recognize that this hypothesis is incom-

patible with any explanation of free fall that relies on the impact of external forces and not just with Descartes' belief in the causal

agency of subtle matter. In fact, Mersenne mentions in this con- nection also another theory which ascribes the cause of free fall to the attraction exerted by the earth.7'

But precisely this theory of terrestrial attraction had stood at the center of Gassendi's recent Epistolae duae de motu impresso a motore translato of 1642, with which Mersenne was not only well ac-

quainted, but whose reading he recommended to numerous cor-

respondents. We must therefore assume that Mersenne, in the passage we have just cited, was also trying to think through the

75 Ibid., 44-45: "Cum dixi filum descendens vel ascendens transire per omnes gradus tarditatis, nolim id assertum existimes vti demonstratum, sed tantum vt probabile, et Galilaeo visum, quandoquidem egregij Philosophi et Geometrae negant illud, et grauia certum habere possunt tarditatis gradum quo incipiant moueri circa centrum, vti fieri probabile, si per terrae fiat attractionem lapidum et aliorum grauium, descensus, aut per expulsionem, vt quemadmodum aqua expellit leuiora corpora, ita grauiora expellat aer, aut alia materia aere subtilior, quae circumactu suo, revolutioneque perpetua lapides et id genus excutiat, aut impellat. Quod tamen non impedit quin grauia suam velocitatem in ratione duplicata temporum adeo proxime augeant, vt sensus nil contrarium in observa- tionibus deprehendat."

76 The possibility that bodies are drawn downward by the earth was also granted by Mersenne's Cogitata, tractatus mechanicus, 21: "Supponamus igitur, quod multi censent probabile, gravitatem corporum nil aliud esse quam terrae tractionem, sive mutuam, qualis est inter magnetem et ferrum, sive terrae solius."




mathematical consequences of Gassendi's dynamics, to which we must now turn our own attention.

A causal foundation for Galileo's kinematics: Pierre Gassendi's

'Epistolae de Motu' and 'de Proportione'

As we now turn to Gassendi, we must of course state, first of all, that this author was, together with Mersenne, the driving force behind the divulgation of the Galilean science of motion in France and one of the chief protagonists in the intellectual debate that Galluzzi has called the "seconde affaire galileenne." In this long, if informal trial, the new science of motion played the role of the defendant, the two Jesuits Fabri and Cazre acted as plaintiffs, and Gassendi as the chief advocate for the defense, soon to be followed

by Le Tenneur, Huygens, Torricelli and others who were all called

upon by Mersenne to participate in the debate.77 But as we com-

pare the roles of the two protagonists of this predominantly French debate, we are struck by the divergence in the development of their respective viewpoints. While both began as brothers in arms in their defense of the Galilean science of motion, Mersenne grew-as we are trying to show in this article-ever more doubtful about the possibility of reconciling it with a mechanistic explanation of gravity. Gassendi, by contrast, remained

throughout his life convinced of the validity of Galileo's law and tried to develop a mechanistic model from which it could be derived. But as we shall try to document in the following pages, there existed some difficulties in all of Gassendi's explanations that were

persistent and indeed inherent and which must have nurtured Mersenne's growing scepticism even further.

The history of Gassendi's public involvement with this issue must begin in the month of October, 1640, as our philosopher left the port of Marseilles on a trireme with the intention of verifying on the open sea the result of a thought experiment described by Galileo in his Dialogo.78 In the presence of Louis de Valois, gover-

" P. Galluzzi, "Gassendi e l'affaire Galilee delle leggi del moto, " Giornale critico della filosofia italiana, 72 (1993), 86-119.

781t is worth recalling that in his Lettera a Francesco Ingoli in risposta alla

"Disputatio de situ et quiete Terrae" (1624), Galileo claims to have carried out the experience on the ship in person (Cf. G.G, 6: 545). For a discussion of this claim see P. Ariotti, "From the Top to the Foot of a Mast on a Moving Ship," Annals of Science, 28 (1972), 191-203, esp. 201-202; L. Conti, "La dimensione sperimentale




nor of the Provence and sponsor of the expedition, Gassendi verified that a heavy ball dropped from the masthead arrived precisely at the foot of the mast irrespectively of whether the ship was at rest or was moving at high speed. This observation carried particularly important implications, for it falsified one of the principal objections usually put forward against the Copernican theory. It showed, by analogy, that from an observation of the behavior of objects placed on the surface of the earth, one could not infer whether the latter was in a state of rest or instead moving.79

In December of the same year, Gassendi wrote two letters to his friend Pierre Dupuy which were published at Paris two years later. Their objective was to furnish a detailed report on the experiments carried out at sea and to analyze their implications for the domains of mechanics and cosmology.80 The central part of the first Epis- tola was, however, dedicated to a causal account of the motion of free fall. After having described the trajectory of a body falling down the mast of a moving ship as a half-parabola, Gassendi went on to explain this figure as the resultant of the combination of two motions, namely of a uniform, rectilinear, and horizontal motion

"impressed" on the object by the moving ship, and of a uniformly accelerated and vertically downward motion which was caused by an external principle. But what could be the nature of the latter?

Although there are various ways by which an external cause can move, it is yet clear that all of them belong to two, as it were, outstanding types, namely impulse and attraction. Let us therefore discover, whether the cause of the perpendicular motion of falling bodies is impelling or attractive or some combination of an impelling and an attractive cause.8'

della relativith galileiana," in C. Vinti (ed.), Alexandre Koyre. L'avventura intellettuale, (Naples, 1994), 549-576; C.R. Palmerino, Atomi, meccanica, cosmologia. Le lettere galileiane di Pierre Gassendi (Ph.D. thesis, Florence, 1998), 9.

79 On Gassendi's experiment, cf. A.G. Debus, "Pierre Gassendi and his "Scien- tific Expedition" of 1640," Archives internationales d'histoire des sciences, 16 (1963), 129-142; Ariotti, "From the Top," passim.

80 Regarding Gassendi's Epistolae de motu, cf. A. Koyr&, Etudes galilennes, (Paris, 1939), 304-317; J.T. Clark, "Pierre Gassendi and the Physics of Galileo," Isis, 54 (1963), 352-370; H. Jones "Gassendi's Defence of Galileo: the Politics of Discre- tion," in Acta Conventus Neo-Latini Guelpherbytani, Proceedings of the Sixth Interna- tional Congress of Neo-Latin Studies (New York, 1988), 221-232; Galluzzi, "Gassendi," 87-89; C.R. Palmerino, Atomi, 7-149.

81 P.G., 3: 489b-490a: "Caeterum, cum plures sint modi, quibus causa externa movet, constat tamen omneis ad duos, tanquam praecipuos pertinere, impul- sionem et attractionem. Age itaque experiamur, an-non motus rerum cadentium, sive perpendicularis, aliqua esset possit causa seu impellens, seu attrahens, seu

potius impellens, et attrahens simul."




Gassendi's answer lay in the hypothesis that the motion of fall was jointly caused by two forces, one of which pushed the body from above while the other pulled from below. The first, which he called vis impellens, was exercized by the air which, when pushed aside by the falling body, was forced to rush upward to fill the

space that had just been evacuated by the body and thereby produced on it an additional pressure a tergo. The vis attrahens, in turn resided in the earth which continuously emitted magnetic particles which formed veritable little chains capable of reaching and catch-

ing far-away objects and of carrying them back down to earth.82 But when Gassendi tries to illustrate the manner in which the

vis attrahens (or the vis impellens, given that the two forces work

analogously)8" makes the body accelerate, he uses a physical ex-

ample which resembles the one frequently invoked in Descartes' letters to Mersenne. For he imagines a globe with an absolutely smooth surface, placed on a perfectly polished plane and put into motion by a little manual push. After having let the ball run for a while with unaltered speed, the same hand will give it a further

push of equal intensity and thereby communicate a second im-

pulse in the same direction. A third push is assumed to follow, and then a fourth, etc. All of them are taken to constitute ictus consimiles, or strokes of equal intensity. The idea is that in this example the globe will increase its speed according to the series of natural numbers. But despite the similarity of Gassendi's model to Descartes', the presumed result of his mental experiment does not

agree with Descartes' explanation of the acceleration of falling

82 Gad Freudenthal has correctly observed that Gassendi does not succeed at all in explaining how it is possible that particles "issuing from a body, conceivably bring about a motion toward it," cf. G. Freudenthal, "Clandestine Stoic Concepts in Mechanical Philosophy: the Problem of Electrical Attraction," in J.V. Field, F.A.J.L. James (eds.), Renaissance and Revolution. Humanists, Scholars, Craftsmen and Natural Philosophers in Early Modern Europe (Cambridge, 1993), 161-172. In 1675, J.B. La Grange had already drawn attention to the contradictory nature of Gassendi's analysis of the mechanism of attraction: "I1 n'est pas difficile de combattre l'opinion de Gassendi touchant la pesanteur, puisqu'apres avoir dit plusieurs fois que la pesanteur consiste en ce qu'il sort perpetuellement de la terre des corpuscules crochus semblables 'a des petits hameCons, lesquels attirent en bas tous les corps qu'ils rencontrent, [...] il avoue luy mesme [...] qu'il ne voit point comment est-ce que ces corpuscules pourraient obliger les corps [...] de descendre [...]. En effet [...] il faut encore qu'il y ait quelque chose qui retire ces petites chaines, ou qui repousse fortement en bas les mesmes corpuscules, apres qu'ils se sont attach6s aux corps pesants," cf. J.B. La Grange, Les Principes de la philosophie, contre les nouveaux Philosophes Descartes, Rouhault, Regius, Gassendi, Le P. Maignan, etc. (Paris, 1675), 192.

83 Cf. P.G., 3: 497a.




bodies. For, as we have seen, Descartes believed neither that each successive push of subtle matter could produce an identical increase of speed in a falling body (given that the faster the body descended, the less it could be accelerated), nor therefore that the

speed of fall could be augmented indefinitely. Furthermore, Gassendi's example is profoundly ambiguous, for

the author does not explain exactly what he intends by ictus consimiles. According to the principle of the relativity of motion which Gassendi himself clearly formulated in his Epistolae de motu,84 the hand can only add a new degree of velocity to the ball with each successive ictus if it remains in a state of rest with respect to the ball-which means, in this case, that it indefinitely accelerates

along with the ball. But if this is indeed Gassendi's intention, then the pitfall of the model lies in the fact that it leads to a sort of

explanatory regressus ad infinitum. For to explain the indefinite acceleration of a motum one needs first to assume an indefinite acceleration of the movens.

In this context, it is worth recalling that precisely this problem had been at the center of an exchange between Galileo and Cavalieri a number of years earlier. In February 1635, Bonaventura Cavalieri sent a letter to Galileo in which he mentioned having invented a contrivance capable of accelerating a small wheel to

infinity by means of a large wheel which turned at constant speed and conveyed to the first wheel successive impulses of equal inten-

sity.85 But Galileo quickly convinced him of the impossibility of his

undertaking, that is of obtaining "with these wheels what happens in the case of free fall of bodies." Galileo's refutation of Cavalieri's model relied on a simple consideration: given that it was impossible that the speed of the small wheel could come to exceed that of the large wheel, the only way to increase the speed of the former was to increase the speed of the latter. And since this was not foreseen in the model, there was no reason for expecting that successive impulses imparted to a body by a motor that remained

"constantly in a given degree of speed" should produce a cumula- tive effect.86

84 Cf. P.G., 3: 478b. 85 G.G., 26: 204-205. 86 Cf. ibid., 230-231. Cavalieri's "speed accumulator" has been brought to my

attention by a recent article by Michel Blay and Egidio Festa. The two authors believe that Cavalieri's letter, in which the "accumulator" is discussed, contradicts what the same author had maintained in his Specchio Ustorio and in his Geometria indivisibilibus: "L'application suggerde par Cavalieri [...] montrait clairement que




Does it not seem as if the same objection is equally valid with

respect to Gassendi's attempt to explain the cause of acceleration in fall through his thought experiment of the hand pushing a ball on a plane? The idea that a ball rolling on a surface or a stone

falling downwards can be indefinitely accelerated by a repetition of ictus consimiles inflicted upon it by a constant force is absurd. As Galileo pointed out to Cavalieri, such an acceleration would re-

quire an increase in the speed of the movens: "if the hand wants to

convey a greater speed" to a sphere-or, mutatis mutandis, the little

magnetic chains to the stone -, "it is first necessary that the hand itself have that greater speed, that is, that it begins to accelerate first, [...] and not that it remains constant in a given degree of

speed."87 Now, leaving aside our own doubts, Gassendi himself did not

think that the ball of his thought experiment would accelerate

according to the odd-number law, for he thought that such an acceleration would require the collaboration of two causal agents.

For assume that that there is one single cause, for example attraction. You will understand that from what we have said, it follows that, since the magnetic rays, like capturing chains, impress upon the stone through contact a motion or impetus, and that they impress it in the first moment such that it does not get deleted, but is conserved in the second moment, in which an-

la grandeur vitesse globale tait conCue comme la "somme" des degres de vitesse successifs, chacun apparaissant a chaque interval de temps. [...] Tout se passait comme si Cavalieri-physicien acceptait ce que Cavalieri-geomitre n'avait pas ose admettre, c'est-a-dire la composition du continu '

partir de ses seuls indivisibles. Ainsi les doutes jetes sur la possibilite d'effectuer la somme d'un nombre infini de degris de vitesse, tombaient lorsqu'il s'agissait de degres attaches de maniere visible, si l'on peut dire, a un corps en mouvement. [...] Ainsi, pour Cavalieri, la vitesse globale, et l'espace egalement, puisqu'il etait cense croitre comme la vitesse, risultaient bien d'une somme que les mathematiques traditionnelles ne permet- taient pas d'effectuer" (M. Blay, E. Festa, Mouvement, 85). Though having discussed at length with the authors, I am still unpersuaded by their interpretation. It does not seem to me as if there was in fact a conflict between the "two Cavalieris," for the acceleration that he wished to obtain by means of his "accumulator" was the result of an addition of homogeneous magnitudes (the degrees of speed)-and this was certainly no procedure that the "mathematiques tra- ditionelles" would have barred. The only step that the conclusions of Cavalieri's Geometria indivisibilibus did not permit was the consideration of the total finite speed of a body (which was represented by a surface) as the sum of an infinite number of degrees of speed (which were represented by lines, i.e. the indivisibles of a surface). But none of his conclusions prevented Cavalieri from assuming that a body could reach in an infinite time an infinite degree of speed thanks to its acquisition of an infinite number of finite degrees of speeds, which were summed up into an infinite line. But it is precisely this assumption that lies at the basis of Cavalieri's letter to Galileo concerning the "accumulator."

87 G.G., 16: 231.




other similar impetus is added [...], in such a way that thanks to this continuous addition, the impetus grows continuously and the speed becomes ever faster. It is certainly easy to predict that from this addition will follow an increase of speed according to the series of natural numbers, so that in the first moment there will be one degree of speed, in the second two, in the third three, in the fourth four."

It is noteworthy that in this passage Gassendi does not draw any distinction between the increase, respectively, of the degrees of

speed, of the total speed, and of the spaces traversed. While Galileo had established in the first corollary to the Theorema I de motu naturaliter accelerato that "while [...] the degrees of speed increase according to the simple series of numbers in equal times, the spaces traversed in these same times grow according to the series of odd numbers starting from unity,"89 Gassendi now believed instead that the spaces traversed in successive but equal moments of time had to grow according to the ratio of natural numbers, just like the degrees of speed themselves. This un- Galilean identification of total speed, degrees of speed, and spaces is, however, not a mathematical mistake as has at times been maintained,90 but appears absolutely correct if one looks at the physical premises on which it is erected. For according to the hypothesis of this French philosopher, the acceleration of bodies is the

product of the action of a force which operates by means of consecutive impulses:

When I speak of 'first moment' I mean the minimum, in which one simple ictus is impressed by attraction, and a minimum space is traversed with a

simple motion, and to which subsequently degrees of speed can be added by repeated ictus.91

8 P.G., 3: 497a: "Nam fac unicam esse causam, exempli gratia attractionem; concipies quidem ex dictis sequi, ut quia radij magnetici, quasi stringentes chor- dulae, contin<g>entem motum, sive impetum lapidi imprimunt, talem imprimant in primo momento, qui non deleatur, sed perseveret in secundo, in quo alius similis imprimitur, qui in priori iunctus perseveret una cum illo in tertio; in quo alius similis adiungitur, atque ita consequenter; adeo ut impetus ex continua illa adiectione continuo increscat, motusque semper velocior fiat. Verum facile erit pervidere consequi ex hac adiectione incrementuum celeritas secundum uni- tatum seriem; nempe ita ut in primo momento sit unus velocitatis gradus, in secundo sint duo, in tertio tres, in quarto quatuor."

89 G.G., 8: 212: "[...] dum [...] gradus velocitatis augentur iuxta seriem simpli- cem numerorum in temporibus aequalibus, spatia peracta iisdem temporibus incrementa suscipiunt iuxta seriem numerorum imparium ab unitate [...]."

90 Cf. Clark, "Pierre Gassendi," 364; E. Festa, "Gassendi interprete di Cava- lieri, " Giornale critico della filosofia italiana, 71 (1992), 289-300, esp. 229-230; Gal- luzzi, "Gassendi," 97.

91 P.G., 3: 497b: "Cum primum momentum accipio minimum intelligo, in quo unus et simplex ictus per attractionem imprimatur, peragaturque minimum spa-




The first moment of the motion of fall is therefore constituted by the interval of time during which the body traverses a minimum of space with a uniform speed which was imparted to it by the first and simple impulse of attraction. To this first impulse are succes-

sively added all others. Each successive push allows the body to

pass through one more minimum of space than it had done in the

preceding moment. This means that the body has two degrees of

speed and passes through two spaces in the second moment; has three degrees of speed and passes through three spaces in the third moment, and so forth. Gassendi's error lies thus not in his calculation, but only in the definition he offers of the property of the motion of fall. For he calls the increase of speed produced in the body by the vis attrahens "continuous,""92 forgetting that by "continuous" must be meant, in the words of Salviati from the Dialogo, an acceleration that "is made continuously from moment to moment, and not discretely from one extended part of time to an-

other."'93 We recall that Galileo considered the ambient air an obstacle

from which one had to abstract so as to arrive at the formulation of the law of free fall. Gassendi, by contrast, believes that only by assuming the presence of a medium capable of pushing the fall-

ing body could Galileo's law be confirmed." His hypothesis is therefore that the vis attrahens initiates the motion of fall, but that from the second moment onward, the vis impellens also comes into

play by giving successive impulses to the body, each of which con- fers to it an additional degree of speed per moment of time. So as to demonstrate more clearly the collaboration between the vis attrahens and the vis impellens, Gassendi makes use of figure 10,

tium, motu exsistente simplici, et cui deinceps accedere, ex repetitis ictibus, gradus celeritatis possint."

92 Cf. P.G., 3: 497a.

"9 Cf. Galilei, Dialogue, 228-229 (= G. G., 7: 255); I have modified Drake's translation.

94 In the first day of the Discorsi, Galileo had deduced the fact that bodies of different specific weight fell in the void with equal speed from their behavior in media of various densities. The argument is roughly as follows: since it can be easily observed that bodies of different weights differ increasingly less in their respective speeds the more the medium in which they descend is supple, it seems that one can conclude that in the void, all of them will fall with the same speed (G.G., 8: 117). The analysis of the different behavior of the same body in air or water serves Gassendi, by contrast, to show that the medium, with its vis impellens, fulfills an active role as a causa adiuvans et adaugens motus deorsum velocitatem.




A

F1

O NO M

JC

Fig. 10: P.G., 3: 498a

where the lines AB and AC stand for the "equally flowing time," the space enclosed by them represent the "uniformly growing speed," and the many triangles which are identical to ADE mean "as many degrees of speed, and hence parts of

space, which the falling body traverses."95

It will suffice to count these

triangles to understand that a mobile which in the first moment had received one degree of speed will receive three de-

grees in the second moment, five in the third moment, seven in the fourth, thus following the series of odd numbers ab unitate. But

although through his joining of forces Gassendi had managed to arrive at the same law of acceleration as Galileo, the figure which he uses to illustrate this law is different than the one found in both the Dialogo and the Discorsi. Gassendi in fact no longer identifies the growth of speed with the surface of a rectangular triangle, but with that of a isosceles triangle. The difference between these two models does not reside only in the choice of "signifiers," but above all in the "signified." While Galileo had the bases of the triangles to symbolize the degrees of speed, Gassendi's corresponding segments KO, ON, NM, ML are devoid of any meaning; the degrees of speed are, by contrast, represented by the triangles ADE, HKO, ILM, etc., whose number corresponds to that of the spaces traversed.

Figure 11, which is used in the Discorsi to illustrate the law of odd numbers formulated in the first corollary of the Theorema II de motu naturaliter accelerato helps, I think, much better than the one used in Gassendi's De motu to understand why the vis impellens and the vis attrahens must together produce an acceleration according to the series of odd numbers ab unitate. The rectangle ADEC, which is equal in surface to the triangle ABC, symbolizes the total

speed of a body which moves in the time segment AC with a constant degree of speed AD. AD is half of the degree of speed BC which is

95 P. G., 3: 498a.




reached in the same time AC by a body which falls with a uniformly accelerated motion.

On the basis of theorem 1, proposition 1,96 we can affirm "without controversies"-in

Sagredo's words-that when the total speeds of the two bodies are identical, then the spaces traversed by them in the time AC will also be identical. Now, the figure shows clearly that if the continuously accelerating body passes, in a second interval of time CI, through a

space that is thrice as large as that traversed in the time interval AC, reaching in the instant I a degree of speed twice as large as it had in the instant C, then a body in uniform rectilinear motion which in the first minimum of time AC had traversed a minimum of

A

> / l G lit

P R , o0 ? 0

Fig. 11: G.G., 8: 211

space with the "simple" degree of speed AD will require, for the second minimum of time, three degrees of velocity so as to traverse three minima of space. Gassendi's error lies thus in his wish to represent acceleration graphically as continuous, though he had defined and analyzed it as discrete. If instead of the area

uniformiter increscens of the triangle, he had used the area difformiter increscens of the "indented" polygon, his readers would undoubt-

edly have found it easier to visualize the effects brought about in the body ex repetutis ictibus of the vis attrahens and the vis

impellens. From what we have said so far, one will be able to understand

why the reading of Gassendi's De motu persuaded Mersenne that neither an explanation of gravity in terms of a vis attrahens nor one based on a vis impellens (be it the air or the subtle matter) were

compatible with the notion that a falling body passed through infinite degrees of velocity. As we have seen, both forces were described by Gassendi as acting in such a way that the speed of the

falling body increased uno initanti at the beginning of each interval of time and that it remained constant until the beginning of the subsequent instant, when it received once more a new in-

96 G.G., 8: 208: "Tempus in quo aliquod spatium a mobili conficitur latione ex quiete uniformiter accelerata, est aequale tempori in quo idem spatium confice- retur ab eodem mobili motu aequabili delato, cuius velocitatis gradus subduplus sit ad summum ed ultimum gradum velocitatis prioris motus uniformiter acce- lerati."




crease. Mersenne must have been aware that the double-distance rule which he had used in his Harmonie universelle to justify the odd-number law was valid in the case of continuous acceleration but not in the case of an acceleration 'by jumps'.

But let us take note of the fact that in his Epistolae de proportione qua gravia decidentia accelerantur (1646), which he had composed to defend the Galilean theory of acceleration against Pierre Le Cazre's attacks, Gassendi was to modify his causal explanation of free fall quite substantially.97 He would now argue that one single and continuously acting force was sufficient to produce the desired increase of speed according to the series of odd numbers. Refer-

ring back to his Epistola de motu, he now admitted:

The mistake lay in the fact that I unwisely admitted that the speeds were in the same proportion as the spaces. For as I did not pay sufficient attention to the fact that the degree of speed acquired in the first moment remained intact in the second moment and was therefore sufficient to cover two spaces, I thought of it as if it could only suffice to cover one single space. Thus when I saw that in the second moment three spaces got traversed, I believed that one space was traversed thanks to the degree that had been conserved while the others were traversed thanks to the other two degrees that had in the meantime been acquired.98

In his Epistolae de proportione, Gassendi uses once more the tri-

angle of speed he had earlier used in his letter to Pierre Dupuy, but he now endows it with new meaning. The segments into which the lines AB and AC are divided (cf. fig. 10) still represent the

equal moments of time which flow ab initio, and the small triangles which make up the large triangle AKL continue to represent the

equal intervals of space traversed by the falling body from its original position of rest. The difference consists in the fact that the

degrees of speed are no longer indicated by the surfaces, but by the bases of these triangles. Gassendi, when giving the reasons

97 The first Epistola de proportione was a response to Cazre's Physica demonstratio qua ratio, mensura, modus, ac potentia, accelerationis motus in naturali descensu gravium determinantur adversus nuper excogitatam a Galilaeo Galilaei Florentino Philosopho ac Mathematico de eodem motu pseudo-scientiam (Paris, 1645); while the second was a reply to Cazre's Vindiciae demonstrationis physicae de proportione qua gravia decidentia accelerantur (Paris, 1645).

98 P.G., 3: 621b: "Iam lapsus fuit quatenus proinde velocitates ut spatia habere se admisi imprudens. Quia enim non satis attendi velocitatis gradum primo momento acquisitum ita integrum manere in secundo, ut ad superandum duo spatia valeret, ipsumque ita habui, quasi solum valeret ad superandum unicum; ideo cum viderem secundo momento tria superari spatia, existimavi facile ita unum superari per gradum manentem, ut duo alia deberent per duos alios, in- terim acquisitos superari."




behind this change, explains that the first interval of time AE "is not an indivisible entity, but can be divided into so many instants or 'timelets' as exist points or particles in AE (or AD)," and that the velocity "grows from the beginning throughout the entire first time, and can be represented by as many lines as the parallels to DE that can be drawn between the points of the lines AD and

AE."99 Gassendi thus admitted that in his Epistolae de motu, he had

supported the hypothesis that the speed of a body grew uno instanti at the beginning of each interval of time and would be conserved intact until the beginning of the successive interval. In the meantime, however, he had persuaded himself that bodies accelerated continuously within each interval and that for this reason the degree of speed reached at the end of an interval of time was sufficient to traverse a space twice as large in the following interval.

This development is not without some irony. For Cazre, who had tried to convince Gassendi with his Physica demonstratio to withdraw his support for the Galilean theory of acceleration, had in fact

produced the contrary effect: Gassendi's reply demonstrates that this author had instead moved more closely to the original Galilean position. While rebutting the arguments of Cazre, who had based his own theory of motion on the assumption that the

speed of fall grew proportionally to space, Gassendi became aware of the following: i) that only the hypothesis of a proportionality between speed and time was compatible with the notion of a continuous acceleration; and ii) that in a continuous acceleration, each degree of speed acquired by the body during a determinate interval of time is preserved unchanged in the successive interval, where "it acts twice as much because of its constancy."'00

The eventual consequence of these new insights was that Gassendi understood that he could simplify his causal explanation of the motion of free fall with respect to the models he had devel-

oped earlier. He now felt that it was enough to entrust the entire effect to the vis attrahens. The resulting theory was of course much more economical and coherent than the one he had developed in his letter to Pierre Dupuy. For the hypothesis that the air exercized a propelling force on the falling bodies had not only rendered the

99 Ibid., 566a '00 Ibid., 608b.




correspondence between downward acceleration and upward deceleration highly problematic,"' but it had also introduced, in his De motu, a radical break between a mechanics of the plenum and a mechanics of the vacuum: for according to this treatise, heavy bodies should have obeyed the Galilean laws of acceleration only in a

plenum; but at the same time, uniform rectilinear motion could

only have been preserved in a vacuum! In the Epistolae de proportione, these two mechanics were finally reunited. There, the medium was no longer considered a quantifiable force, but an obstacle from which one had to abstract so as to be able to give a mathematical description of the phenomenon of free fall.

The publication of Gassendi's Epistolae de proportione (1646) did, however, not persuade Mersenne to abandon the scepticism, first

expressed in the Cogitata (1644), regarding the possibility of rec-

onciling the odd-number law with an explanation of gravity in terms of terrestrial attraction. As we shall see further below, in his Novarum observationum... tomus III Mersenne was to praise the skill with which Gassendi had refuted Cazre's theories, but at the same time expressed once more his own conviction that neither some vis attrahens nor any vis impellens could make falling bodies accelerate according to Galileo's law.

According to Paolo Galluzzi, the reason for Mersenne's overt

scepticism is due to the fact that Gassendi's Epistolae de motu (1642) and de proportione (1646), respectively, had forged a strong alliance between Galilean dynamics and Copernican cosmology. Galluzzi surmises that this combination might have frightened Mersenne, pushing him "to formulate also for the laws of motion a kind of 'Osiander argument' [...], to the great satisfaction of the Fathers of the Society of Jesus."102 Though such a motivation should of course not be excluded, and was certainly at work in the case of a number of Aristotelian foes of Galileo's new science of motion, I believe that Mersenne's growing scepticism must be explained much rather as a consequence of a number of theory-inherent problems. For though the causal explanation of acceleration furnished by Gassendi in his De proportione presented itself as a theory

101 In his last letter to Cazre, Gassendi himself admits that if the air were able to exercize a tergo some pressure on the falling bodies, the same would have had to be the case for ascending bodies. P.G., 3: 622a: "Non ne ergo, si descendendo iuvat air impetum a gravitate impressum: necesse est, ut ascendendo iuvet

impetum impressum a manu?" 102 Galluzzi, "Gassendi," 112.




fully compatible with Galileo's account, it was beset by mathematical and physical difficulties. Gassendi did maintain, it is true, that the acceleration of fall occurred in a continuous manner and represented the degrees of speed by lines and no longer by surfaces. But he could still not be moved to declare explicitly that time was

composed of an infinity of instants, space of an infinity of points, and speed of an infinity of degrees. To admit the mathematically indivisible would have created enormous problems for Gassendi, the atomist, who after all believed that all physical magnitudes must be made up of indivisible, but extended, parts.'03

The theory of acceleration as given in the Epistolae de proportione was ambiguous not only from the mathematical point of view (because of Gassendi's silence on the problem of the passage of the

body through infinite degrees of speed), but also from the physical perspective. How could a force which acted by contact possibly produce a continuous acceleration? We have earlier seen that Gassendi's model of the ball pushed at regular intervals by a hand was in reality incapable of explaining the indefinite acceleration

1os Incidentally, Gassendi's unwillingness to admit the existence of mathematical indivisibles was to lead in the Syntagma to a paradoxical solution. On the one hand, its author reconfirmed his support for the Galilean theory of acceleration as a continuous process, on the other he described the uniform rectilinear motion of the macroscopic res concretae as an alternation of motion and rest. In the. Syntagma, Gassendi explains that the only really continuous movement found in nature is the rectilinear uniform motion of the atoms which move all at a speed of one minimum of space per minimum of time. The motions of the compound res concretae, which are slower than those of the atoms, are in fact all discontinuous: "Ita licere videtur concipere motum, quo Atomi per inane ferri dicuntur [...] esse velocissimum; omneis vero gradus, qui ex illo, ad meram usque quietem sunt, ex intermistis paucioribus, pluribusve quietis particulis esse." (P.G.,1: 341b). The first to observe the radical inconsistency between the principle of inertia stated by Gassendi in his Epistulae and the theory of discontinuous motion set forth in the Syntagma philosophicum was A. Koyre, "Pierre Gassendi: le savant," in Centre international de synthese, Pierre Gassendi, 1592-1655. Sa vie et son oeuvre (Paris, 1955), 59-70 and 108-115, esp. 109. Koyre's argument has been further developed by P.A. Pav, "Gassendi's Statement of the Principle of Inertia," Isis, 57 (1966), 24- 34; M.H. CarrY, "Pierre Gassendi and the new philosophy," Philosophy, 33 (1958), 112-120; W. Detel, "War Gassendi ein Empirist?," Studia Leibnitiana, 6 (1974), 178- 221; B. Brundell, Pierre Gassendi. From Aristotelianism to a New Natural Philosophy (Dordrecht, 1987), 79. Disagreement with Koyr6's criticism has been voiced by O.R. Bloch, La philosophie de Gassendi. Nominalisme, mate-rialisme et metaphysique (The Hague, 1971), 226-227, who claims that the theory of the discontinuity of motion plays only a passing role in the Syntagma, being nothing else than an ad hoc hypothesis introduced so as to account for the paradoxes of motion. Bloch's interpretation has been convincingly refuted by M. Messeri, Causa e spiegazione. Lafisica di Pierre Gassendi (Milan, 1985), 86-93.




of a body in discrete steps. It should be obvious that it was even less capable of explaining continuous acceleration. As a matter of fact, in the Epistolae de proportione, the reader gets acquainted with both a law of acceleration and the name of the presumed under-

lying cause, that is, terrestrial attraction, but with no physical model to visualize or otherwise imagine the functioning of this cause. The fragile crossbreed between Galileo's law and Gassendi's causal explanation left no space for any physical model-building.

Physical reasons against Galileo's law: Fabri and Baliani

We have seen how Gassendi adjusted the original causal explanations of free fall of his Epistolae de motu so as to obtain a more faith- ful dynamic translation of Galileo's kinematics. For Gassendi, then, Galileo's odd-number law remained the non-negotiable part of his

theory; his attempts to lessen the evident tension between mathematics and causal explanation were thus directed exclusively at a redefinition of the models of the physical forces held responsible for gravity and acceleration. But obviously, the conflict between mathematics and physics could also be resolved in the opposite way, by trying to adjust the mathematics to the hypothetical physical models. A figure who well exemplifies the latter reaction is Giovan Battista Baliani, whose name often figures in Mersenne's works.

Baliani is a particularly interesting case as he had published, in the very year of Galileo's Discorsi, a work entitled De motu natura- lium gravium solidorum in which he formulated-quite independently from Galileo, as it would appear-the same law of odd numbers. A few years later, however, and just a few months after the publication of Gassendi's Epistolae de proportione, Baliani published a second edition of his original De motu in which he explicitly rejected his earlier odd-number law and defended instead the law of natural numbers.'04 The reason for this change of mind lay in the fact that he had simply not been able to find any causal ex-

planation that could have accounted for the former law, but more than one explanation to yield the latter. Baliani had now come to

104 For an account of Baliani's theory of free fall, see S. Moscovici, L'experience du mouvement. Jean-Baptiste Baliani disciple et critique de Galilee (Paris, 1967); G. Baroncelli, "Introduzione," in G.B. Baliani, De motu naturali gravium solidorum et

liquidorum, tr. and ed. Baroncelli (Florence, 1998).




believe that a falling body which was moved either naturally or

violently ("seu naturaliter a gravitate deorsum, seu violenter ab impellente") received at regular intervals of space new identical

degrees of impetus. Each consecutive impetus, in turn, was added to the ones previously acquired and remained intact until the end of the movement. The implication of this idea was that when the

impetus doubled, tripled or quadrupled, the velocity of the mobile doubled, tripled or quadrupled also.

Just like Gassendi in the first letter to Pierre Dupuy, Baliani described acceleration as a sum of uniform motions, each of which

possessed a speed vn which was a multiple of the speed v1 of the first motion. But unlike Gassendi's, Baliani's theory had only one force at play, which meant that the law of acceleration had to follow the series of natural numbers. He justified both his previous 'error' and the continued adherence of some contemporaries to the Galilean formulation of the law by saying that the difference between the two laws was experimentally imperceptible.'05

Paolo Galluzzi has given us good reasons for believing that Ba- liani's change of mind did not occur independently ofJesuit pressure.'"6 It has in fact been repeatedly observed'07 that Baliani's later

theory resembled the views of the Jesuit Honore Fabri as presented in the Tractatus physicus de motu locali of 1646.108

We have seen that in the ballistic section of the Cogitata physico- mathematica, Mersenne mentions among others the law of acceleration formulated by the "philosophus subtilissimus" Honore Fabri. We know that the Minim had already in 1643 begun a correspondence with this Jesuit in which he communicated his various

'05 G. B. Baliani, De motu naturali gravium solidorum et liquidorum (Genua, 1646), 71: "Augetur igitur, ni fallor, motus iuxta progressionem arithmeticam, non numerorum imparium, ab unitate huc usque creditam, sed naturalem; at nihilo- minus, cum fere idem effectus subsequantur, ob insensibilem discrepantiam, mirandum non est, creditum fuisse spatia esse in duplicata ratione temporum; quandoquidem etiam si verum precise fortasse non sit, est attamen adeo veritati proximum, ut veritatem in adhibitis experimentis sensus percipere nequiverit, quamobrem excusandi sunt quicunque ita censuerunt."

106 Galluzzi, "Gassendi," 114-115. For Baliani's relations with the Jesuits, cf. C. Costantini, Baliani e i Gesuiti. Annotazioni in margine alla corrispondenza del Baliani con Gio. Luigi Confalonieri e Orazio Grassi (Florence, 1969).

'07 Cf. S. Drake, "Impetus Theory and Quanta of Speed before and after Galileo," Physis, 16 (1974), 47-65, esp. 50-51; Dear, Mersenne, 216; Galluzzi "Gassendi," 115.

108 The Tractatus physicus (Lyon, 1646) was a collection of Fabri's physics lec- tures collected and edited by his student Pierre Mousnier.




doubts regarding the validity of the law of natural numbers. In a long letter of August 1643-which also happens to be the

earliest extant piece of this correspondence-Fabri enumerated a number of Mersenne's objections and suggested answers to each of them. First of all, he tried to account for the discrepancy between his own law of acceleration and the results of experimental measurements by saying that if only he had been able to use "instants" as his units of time, he would have been able to show that the spaces traversed in free fall actually did grow according to the law of natural numbers. But since "in actual experience one can

only take a portion of time that contains various instants, as is certain, one should not be surprised to find that the proportion found by experience corresponds roughly to Galileo's."'19 In other words, while Deschamps had criticized the inflexibility of the law of natural numbers as a fundamental defect,1o? Fabri saw in it a way for saving the phenomena. The larger the time interval chosen as the unit of measure-so he argued-the smaller the difference between his own law of acceleration and Galileo's, the implication being that his own true law would only be completely verified if one could measure the spaces traversed in the physical instants.

The notion of the finite physical instant which constituted one of the essential foundations of Fabri's theory of motion was hard to accept for Mersenne. The composition of the continuum out of indivisible parts not only contradicted the entire book X of Euclid's Elements, but it also did not allow for an explanation of incommensurable ratios between magnitudes. In answering the

109 C.M., 12: 289: "[...] dans l'experience l'on ne peut que prendre une partye qui contient plusieurs instants, ce qui est certain, il ne faut pas s'estonner sy la proportion trouvee par l'experience respond a peu pres a celle de Galilee." An

analysis of this letter is found in Galluzzi, "Gassendi," 100-102. For Fabri's theory of acceleration, cf. Drake, "Impetus," esp. 47-65; id., "Free Fall from Albert of

Saxony to Honore Fabri," Studies in History and Philosophy of Science, 5 (1975), 347- 366; D.C. Lukens, An Aristotelian Response to Galileo: Honore Fabri, S.J. (1608-1688) on the Causal Analysis of Motion, (Ph.D. thesis, University of Toronto, 1979); Dear, Discipline, 138-144. For other aspects of Fabri's science, cf. E. Fellmann, "Die mathematischen Werke von Honoratus Fabry," Physis, 1 (1959), 6-25; A. Boehm, "L'aristotelisme d'Honor6 Fabri (1607-1688)," Revue des sciences religieuses, 39 (1965), 305-360; E. Caruso, "Honore Fabri gesuita e scienziato," Miscellanea seicentesca. Saggi su Descartes, Fabri, White (Milan, 1987), 85-126; E. Fellmann, "Honore Fabri (1607-1688) als Mathematiker-eine Reprise," in P.M. Harman, A. Shapiro (eds.), The Investigation of Difficult Things: Essays on Newton and the

History of the Exact Sciences in Honour of D. T. Whiteside (Cambridge, 1992), 97-112.

•0 Cf. above, pp. 295-96.




objections raised by the Minim, Fabri tried, in a rather unique manner, to reconcile his own physical atomism with the Aristote- lian theory of the continuum by arguing that "being potentially divisible is the same thing as being actually indivisible or not to contain distinct things of which the one can truly be separated from the other, but to be able to correspond to distinct things only by coextension.""' The fact that matter was composed of extended and impenetrable parts of given shapes "is of no weight against the

incommensurability of Euclid, who is not considering matter, but

only its extrinsic quantity or extension."'12 The letter continues with Fabri's attempt to clarify his own physi-

cal explanation of free fall, which had in God the first cause of its motion and in the impetuosity its immediate cause. This immediate cause produced in each instant of time a new point of impetuosity, while the first cause made sure that the points previously ac-

quired were preserved. To Mersenne, who had earlier objected that the conservation of motion was in no need of a cause, Fabri now pointed out that

if it is true [...] that the degree of speed which existed in the first instant remains in the second instant together with the degree that has been added, then it is doubtlessly necessary that its cause remain also, which is its impetuosity. Hence it is necessary that it be conserved by a different cause, for otherwise the second degree would not be produced."1

In the concluding pages of the letter, the Jesuit tried to account for the fact that bodies of different sizes and materials fell with the same speed. He argued that the spatial extension of the moving body did not influence "the intention of the effect produced in- side, but only outside" the body. With this, he wished to express that a point of matter and a cannon ball fell in the void with the same speed,"14 but that the effect they each produced upon impact with an external body was directly proportional to their respective weights.

A complete and detailed exposition of the same theory was to

.I C.M., 12: 291. 112 Ibid., 292. "' Ibid., 296: "[...] si c'est vray [...] que le degr6 de vitesse qui estoit au pre-

mier instant demeure encores au 2d instant avec le degr6 qui est adjouxt6, il faut sans doubte que sa cause demeure aussy, qui est l'impetuosit6; donc il faut qu'elle soit conserv6e par une autre cause; autrement le 2d degr6 ne seroit pas produit."

114 Note that one of the crucial points distinguishing Fabri's physics from Aristotle's lies in his admission of the possibility of a motion in the void, cf. ibid., 299.




be published in the Tractatus physicus de motu locali (1646), whose first two books treat de impetu and de motu naturali deorsum, respectively, in a rigidly axiomatic fashion. At the beginning of the first book, Fabri defines "impetus" as a permanent quality, which is

"really separate from the substance of the moving body" and which is the formal cause of motion."' If motion is the secondary formal effect of impetus, impetus in turn is the effect of an efficient cause. Here Fabri introduces a distinction between violent impetus, which is produced by an external force and does not have a determinate direction, and natural impetus, which is the product of the substance of the body, and is always directed towards the center of the earth."6 It is the natural downward motion to provide the subject of the second book of Fabri's Tractatus; the first theorem is dedicated to demonstrating that this motion comes from within the

body ("ab intrinseco"). In his proof, the Jesuit proceeds by elimi- nation, rejecting one after the other all explanations of free fall which appeal to an external force. He himself was convinced that bodies could not be pushed downward either by air, by some

magnetic force, by celestial matter, or by light, nor indeed "by any other extrinsic agent, as can be established by induction.""' Fabri is thus obliged to conclude that the speed of fall increases as the effect of an internal cause which also grows; this internal cause is not gravity "which is always the same," but the impetus, which is the

product of gravity."8 The action of this impetus is described in the Tractatus in terms

that are similar to those we have already found in the letter to Mersenne. In fact, Fabri insists once more that time is made up of indivisible instants in which "the impetus equally grows and is re- inforced.""9 And he explains:

I have used above the expression "in equal instants," because the nature of time cannot be explained other than by finite instants as I will demonstrate in the Metaphysica; whatever it may be, I call an instant that whole time in which something is produced all at once.120

15 Fabri, Tractatus, 17-19. 116 Ibid., 69. 117 Ibid., 76-79. 118 Ibid., 80. 119 Ibid., 87. 120 Ibid., 87-88: "[...] dictum esse supra instantibus aequalibus, quia temporis

natura aliter explicari non potest, quam per instantia finita, ut demonstrabimus in Metaphysica; quidquid sit, voco instans totum illud tempus, quo res aliqua simul producitur [...]."




The demonstration of the existence of physical instants, to which Fabri refers in the lines just quoted, is found in the IX book of the

Metaphysica, which Fabri had Mousnier publish in 1648. Here Fabri

argues that "there are physical instants, because there is action

through which a thing is."'21 The existence of things requires a

conserving action, and this action has to change "in single instants, for a permanent thing existing now might not exist in the follow-

ing instant."'22 But given that nothing can exist or move in a mathematical instant, time must be composed of physical instants in which physical actions take place totae simul.

In the Tractatus, Fabri accordingly defines the first instant of motion as "the whole time in which the first acquired impetus is

produced totus simul."'23 This first instant is followed by equal instants in each of which the total previous impetus is conserved while a new impetus is also acquired. Therefore, if in the first instant there is one degree of impetus in the falling body, there will be two in the second instant, three in the third and so forth, "according to the arithmetical progression."'24 And since the velocity grows just like the impetus itself and the spaces traversed just like the

speed, it follows that "spaces grow in single and equal instants

according to the arithmetical progression" (cf. fig. 12).125

K

Fig. 12: Fabri, Tractatus

physicus, s.p.

But Fabri carefully underlines, once more, that this ratio is only valid for the spaces that are traversed in the physical instants and that if one chooses as one's measure

temporal units composed of various instants, "the ratio between spaces will turn out to be greater than the ratio between

speeds."''26 And given that in experimental settings one always takes units of time that are made up of a huge number of instants, it is obvious that the measured growth of

spaces will have to follow a different progression-and in fact one that is close to

121 H. Fabri, Metaphysica demonstrativa, sive scientia rationum universalium (Lyon, 1648), 371.

122 Ibid., 367. 123 Fabri, Tractatus, 89. 124 Ibid., 88. 125 Ibid.. 126 Ibid., 89.




Galileo's.127 Fabri summarizes the difference between his own law and Galileo's in the following words:

Indeed in that common view, according to which time is made up of parts that are actually infinite, Galileo's progression can take place. This, then, must be considered the crux of the problem: the simple progression has a

physical principle, but no experimental evidence; the progression according to odd numbers has evidential confirmation but no principle. We reconcile both of them with the physical principle and with experience, for the first

progression becomes the second, when one assumes sensible parts of time, and the second turns into the first, when one assumes ultimate instants."28

The fact that Galileo's theory of acceleration does not provide a causal explanation of the phenomenon of fall is in Fabri's eyes a clear sign of its inadequacy. The Jesuit is convinced that the law of natural numbers, which he has derived from a physical analysis of the growth of the impetus, has to be preferred to Galileo's "pro theorica rei veritate," despite the fact that it cannot be verified

experimentally. In the context of his attempted confutation of Galileo's science

of motion, Fabri seems to attribute a particular weight to theorem 61 of his Tractatus, which is designed to prove that:

A naturally accelerated motion does not pass through all degrees of speed. For there are as many degrees of this passage as are the instants that make up the duration of this motion, since in each instant a new impetus is added. But there exist no infinite instants, as we shall prove in the Metaphysics. And even if there were infinite instants, this passage would still not be through all degrees of speed, for there would be some degree of speed that this series of degrees does not comprise.'29

The fact that a body rolling down a plane moves more slowly than one which falls vertically and that the speed of fall is slower in

127 Ibid., 115. 28 Ibid., 108: "immo in communi illa sententia, in qua dicitur tempus constare

ex partibus actu infinitis, progressio Galilei tantum locum habere potest; igitur haec esto clavis huius difficultatis; progressio simplex principium physicum habet, non experimentum; progressio numerorum imparium experimentum non

principium; utramque cum principio et experimento componimus; prima enim si assumantur partes temporis sensibiles transit in secundam, secunda in primam, si ultima assumantur instantia."

129 Ibid., 96: "Motus naturaliter accelerato non propagatur per omnes tarditatis gradus; quia tot sunt huius propagationis gradus, quot sunt instantia, quibus durat hic motus, cum singulis instantibus nova fiat impetus accessio, sed non sunt infinita instantia, ut demonstrabimus in Metaphysica; praeterea licet essent infinita instantia, non fieret adhuc per omnes tarditatis gradus haec propagatio; quia daretur

aliquis gradus tarditatis, quem non comprehenderet haec graduum series; [...]."




water than in air proves in Fabri's eyes that these various motions must begin each with different initial speeds. This signifies, for him, that even if there do exist infinite instants of time, it would be physically impossible that a body actually passes through an infinite number of degrees of speed.

In this section, we have seen that for Fabri, the passage of the mobile through infinite degrees of speed was the crux of the entire problem of the fall of bodies. His arguments helped to rein- force Mersenne's conviction that this was indeed the issue that had to be settled before any law of acceleration could be accepted as

physically valid. As we shall see in the following section, his interest in this question grew to the point where he would call upon a number of contemporary natural philosophers and mathematicians to propose solutions.

Mathematical reasons in favor of Galileo's law: Le Tenneur, Torricelli, and Huygens

Soon after the publication of the Tractatus physicus de motu locali, Mersenne had called on the mathematician Jacques Alexandre Le Tenneur to defend the Galilean theory of acceleration against the attacks of the "most acute" Fabri.3so In April 1647, the Minim received a very long letter in return in which Le Tenneur attempted to falsify the hypothesis of a discontinuous growth of speed to-

gether with the law of natural numbers.'3' In his criticism of Fabri's

theory, the Parisian mathematician insisted in particular on the

contradictory nature of the concept of physical instant. Le Ten- neur recalled that Fabri had rejected Galileo's view because it

required the existence of mathematical instants. But his own

impetus theory relied on nothing else but such mathematical instants! For how else was one to interpret the idea that the moving body acquired at the beginning of each physical instant of time one additional degree of speed totus simul and kept it intact while tra-

versing the minimum? What else than a mathematical indivisible was this "initium" of the physical minimum? The fact that in the Tractatus physicus, the growth of the spaces covered by a falling body was represented non per triangula, sed per rectangula (cf. fig.

130 Cf. Le Tenneur's letter to Gassendi of 16 January, 1647, C.M., 15: 49. 131 Le Tenneur to Mersenne, 13 April, 1647, C.M., 15: 173-199




12) showed furthermore clearly that it was Fabri, and not Galileo, to assume that acceleration took place in one mathematical instant and that it was in one mathematical instant that the body jumped from a state of rest to a first degree of speed, and then to a second, third, etc.'32 Finally, the overall superiority of Galileo's law of acceleration over Fabri's was also demonstrated, in the eyes of Le Tenneur, by the fact that only the former was valid irrespective of whatever unit of time was chosen for the measurement of the first interval of time. Like Deschamps before him, Le Tenneur recognized the property nowadays called "scalar invariance" and

pointed out that

the multiplication of times according to any proportion whatsoever always confirms the uniform proportion among the spaces, and it does not happen that you get a larger or a smaller space if the equal times get longer or shorter [...]. Nor will there be a larger ratio between four spaces and two spaces, than between two spaces and one space; nor will there occur a larger or smaller space. [...]. But all is found to cohere and agree marvelously.'"s

A

D E

F G

2a

Fig. 13: C.M., 15:

197

In the concluding pages of his letter, Le Tenneur discussed in detail some of the arguments used by Fabri in his attack on Galileo, and in particular theorem 61 of book II. Contrary to Fabri, Le Tenneur was of the view that it was quite possible to imagine that a moving body descending along an inclined plane and another in vertical free fall both passed through an infinity of de-

grees of speed, "although the parts of speed are larger in the inclined plane than in the perpendicular line." As was in fact shown

by figure 13, the infinite number of lines that could be drawn

parallel to the base BC divided the cathetus AB and the hypot- enuse AC in parts of different sizes but equal in number.'M

While Le Tenneur had no qualms about admitting that in violent or artificial motion, a body could in fact jump immediately from rest to a determinate degree of speed, he excluded that such was the case with naturally accelerated motion.

In the Novarum observationum tomus III, Mersenne writes that Le

i32 Ibid., 178-182. 133 Ibid., 195: "[...]multiplicatis in qualibet proportione temporibus con-

firmatur semper uniformis spatiorum proportio, nec oritur unquam spatium maius aut minus vero sive augeantur tempora aequalia, sive minuantur [...]. Nec invenietur maior ratio quatuor spatiorum ad duo quam duorum ad unum nec orietur spatium minus aut maius vero: [...], sed omnia mirum in modum sibi invicem cohaerere et consentire reperiuntur."

134 Interestingly enough, Le Tenneur's reasoning is absolutely analogous to that used by Galileo in an early Latin draft of a theorem of the Discorsi, which has been analyzed in P. Galluzzi, Momento. Studi galileiani (Rome, 1979), 360.




Tenneur had claimed that the odd-number law could be true even if the descending body did not pass through infinite degrees of

speed.'35 While Mersenne's assessment may at first sight look mis- taken, a closer look at Le Tenneur's De motu naturaliter accelerato of 1649 (in which his letter to Mersenne was included) will clarify this claim.

In the third part of his De motu, which rebuts all objections com-

monly employed against Galileo, Le Tenneur writes that the Florentine mathematician had only managed to prove that a fall-

ing body passed thorough infinite degrees of acceleration, but not that it passed through infinite degrees of speed. For should it really be the case, as Cazre had suggested, that all bodies have a "innate

speed of a certain determinate degree" ("nativa celeritas certo

gradu determinata"), this would imply that a projectile thrown

vertically upward from E to A (fig. 14), even if decelerating in a continuous manner,

would yet not pass through all degrees of slowness between point E (from where it is thrown) and point A (where the impressed impetus ceases to be active). But its innate speed AL would remain, and since it is ever divisible to infinity just like any quantity, there would always remain infinite degrees of slowness still to be traversed by the ascending body.'36

Still, Le Tenneur is convinced that each falling body actually does

pass through infinite degrees of speed. He observes that even if bodies possessed a native degree of speed, this would probably consume itself together "with the impressed impetus [...] before the ascent of the upwardly thrust body had ended."''7 To this are added two more arguments which appeal in different ways to the

principles of the simplicity and uniformity of nature. The first ar-

gument is based on the observation that nature usually does not

pass "from one terminus to the other under omission of the

C

E Fig. 14: Le TenneL De motu, s.]

135 Mersenne, Novarum observationum, 132: "[...] Clariss. Tenneurius, amicus singularis, scripto nundum vulgato [ sc. "De motu naturaliter accelerato"] refellit [per numeros naturales progressionem], ut & priorem eruditissima epistola, in qua praesertim illud placet, quod nostram illam per numeros impares progressionem evincit, licet gravia, a quiete casum inchoantia, non transirent per omnes gradus tarditatis."

16 J. A. Le Tenneur, De motu naturaliter accelerato tractatus physico-mathematicus (Paris, 1649), 110: "[...] certe non transibit [...] per omnes tarditatis gradus a puncto proiectionis E, usque ad A, punctum, in quo cessat quidem omnis impetus impressus, at remanet nativa celeritas AL, quae cum sit adhuc divisibilis in infinitum, sicut quaelibet quantitas, remanet adhuc infiniti tarditatis gradus a gravi ascendente pertranseundi."

1'37 Ibid., 111.




middle."'38 The second invokes once more the quality of scalar invariance which only Galileo's law possesses:

It must needs be the case that the first space is to the second space like the two first spaces to the two subsequent ones, as has been shown against Fabri, because we obviously need a principle of uniformity in natural events as these need to proceed in an uninterrupted course. The consequence of this is that heavy bodies have no innate speed, but that in falling, they pass through all degrees of slowness and speed."'9

In sum, mathematical coherence and simplicity were the two cri- teria that in Le Tenneur's eyes indicated the superiority of Galileo's theory of acceleration over those of Cazre and Fabri.

Le Tenneur was not the only one to rush to Galileo's defense. Further letters of support by Torricelli and Huygens soon reached Mersenne's convent. In January of 1645, Mersenne had written to Torricelli explaining that neither Roberval nor Descartes had been convinced by the validity of the "fundamentum Galilaei"-again, the idea that the falling body passed through infinite degrees of

speed from its original position of rest-believing instead that the

B

E D

AC

Fig. 15: C.M., 13: 338

motion of fall initiated with a determinate

degree of speed. Torricelli's answer was very terse. The Ital-

ian stated simply that he did not understand how a heavy body could jump from rest to a certain degree of speed without passing through all intermediate degrees. And he then furnished an additional geometrical argument (fig. 15): for a point proceeding with uniform speed from B towards D it will be necessary, before it reaches a distance

from the cathetus that is equal to DE, to traverse "omnes minoris distantiae gradus usque in infinitum."'140

Mersenne would continue to write occasional letters to Torri- celli in which he formulated physical objections to the Galilean

theory of accelerated motion. But, as Galluzzi has shown, the dis-

138 Ibid., 112. 139 Ibid., 114: "Cum ergo debeat esse necessario, ut primum spatium, ad secun-

dum, ita duo priora simul, ad duo posteriora simul, ut contra Fabrium ostensum est, ut nimirum servetur uniformitas in actionibus naturalibus, quae uno & eodem tenore progredi debent; consequens est gravia nullam habere nativam celeritatem, sed successive transire in descendendo per omnes tarditatis aut velocitatis

gradus." 140 Torricelli to Mersenne, 27 January, 1645: C.M., 13: 338.




ciple of Galileo was highly unwilling to be dragged into the debate.'41

Greater satisfaction for the Minim was to be gotten from the still

very young Christiaan Huygens. Upon hearing from Constantijn Huygens, Sr., that his son had worked out a demonstration of the odd-number law, Mersenne wrote to Christiaan on October 13, 1646, describing all the reasons for which he had come to believe that the foundations of Galileo's science of motion were wobbly. As Dear has observed, these reasons had to do only with physics.'42 Mersenne had persuaded himself that the speed of free fall possessed both a natural maximum and a minimum: on the one hand, it was impossible for a falling body to exceed its natural impetuosity, on the other, it was necessary that its fall began with a determinate speed. And it was this second conviction that led him to re-

ject the odd-number law:

Furthermore, so as to maintain in the vacuum the proportion of odd numbers, it would be necessary that the body passes through all degrees of speed from the beginning of its descent, which doesn't happen though Galileo thought so, for the stone has already a certain speed when it begins its fall.'43

It took Huygens only a few days to respond, and his answer was in favor of the Galilean law of motion. Since he agreed with Mersenne that the odd-number law was based on the hypothesis of the passage of the falling body through infinite degrees of

speed, he put his epistolary efforts into proving this hypothesis by mathematical means; the odd-number law, he tacitly assumed, would follow from this proof.

He invited Mersenne "to concede" that falling bodies had to accelerate in such a manner that the space they traversed in a first interval of time of whatever size stood to the space traversed in a second, identical, interval of time in the same ratio as the space traversed jointly in the first two intervals to the space traversed

'4' Galluzzi, "Gassendi," 104. 142 Dear, Mersenne, 211. '43 Mersenne to Christiaan Huygens, 13 October, 1646, C.M., 14: 539: "D'ai-

Ileurs il faudroit pour garder tousiours in vacuo la proportion des nombres impairs, que le grave tombast par tous les degrez de tardivet6, depuis le commencement de sa cheute, ce qui ne se fait pas quoy qu'aye pens6 Galil6e, car la pierre a desia une certaine vitesse, en commen;ant la cheute." On Mersenne's relation to the young Christiaan Huygens, cf. A. Beaulieu, "Christiaan Huygen et Mer- senne l'inspirateur," in Huygens et la France (Paris, 1981), 25-31. The exchange of letters between Huygens and Mersenne is also discussed in Galluzzi, "Gassendi," 109-110; Dear, Mersenne, 212-214.




jointly in the third and fourth interval. This essential property having been established, Huygens demonstrated that it could not be satisfied by any geometrical progression and hence not even by Cazre's, and that among the arithmetical progressions, only Galileo's, but not Fabri's, was able to satisfy it.'44

Mersenne declared his admiration for the "gentillesse" of

Huygens' demonstration. At the same time, he felt compelled to confess that his scruples had not been completely eliminated. As we shall see in the concluding paragraph, it was not a mathematical proof of the validity of Galileo's law of acceleration that Mersenne was seeking, but a causal explanation from which that law could be deduced. And this was something that neither Le Tenneur nor Torricelli nor indeed Huygens were able to supply him with.

Mersenne's unresolved doubts

Mersenne's doubts regarding the possibility of constructing an exact science of motion found their most detailed expression in the Novarum observationum tomus III of 1647. In the opening lines of chapter 15, which carries the title "Variae cogitationes de casu

gravium iterum expensae," Mersenne confesses his growing per- plexity concerning the validity of the same odd-number law that he had once believed to have verified empirically. He narrates how in recent years this law has been valiantly defended by Gassendi and Le Tenneur who have with cogent arguments confuted both Cazre's and Fabri's alternative laws.'45 But Mersenne adds that the

144 C. Huygens to Mersenne, 28 October, 1646, C.M., 14: 570-572. It must be evident that the argument employed by the young Huygens is very similar to the one we encountered above with Le Tenneur: both authors individuate in the property of scalar invariance the decisive advantage of the Galilean law over the rivaling laws. The only difference between Huygens and Le Tenneur lies, however, in the presentation of this argument. While the latter, when comparing the three laws, had found that Galileo's was the only one to possess this property and therefore seemed to satisfy better the principle of the uniformity of nature, Huygens behaves as if this property was some conditio sine qua non for any true law of free fall and which only Galileo's can satisfy. Given that he does not try to justify in any way whatsoever why this property was essential, his argument has- not unjustly-been accused of circularity by Nardi ("Spazi," 336-337) and P. Costabel ("Huygens et la mecanique. De la chute des corps a la cause de la pesanteur," in Huygens et la France (Paris, 1981), 139-152, esp. 140-141). To my knowledge, no one has so far drawn attention to the resemblance between the arguments of Huygens and Le Tenneur.

145 Mersenne, Novarum observationum, 131-132.




defeat of these rivals is still not equivalent to a victory of the Galilean law. He praises Le Tenneur's still unpublished mathematical argument in favor of the Galilean law, of which he offers a summary, as a very shrewd piece of reasoning, but then goes on to add that it yet

has no power of demonstration [...], just as the reasons, which have so far been adduced in favor of the double motion of the earth, demonstrate noth-

ing, as has been well stated by the Commentator of Aristarchus, even though some people claim that it moves because of the power of the reasons that seem to confirm it.146

The passage seems to involve the following reasoning: Admittedly, the Galilean theory of acceleration is more economical than either Fabri's or Cazre's, in the same way in which the Copernican theory is more economical than the Ptolemaic model. But such a mathematical advantage does not entail the truth of the theory, at least as long as there exists no physical proof of its validity. And just as the rotation of the earth has not yet been demonstrated, so are we still expecting the physical proof of Galileo's law.

These words also signify that Mersenne had evidently not been convinced by Gassendi's attempt to provide a causal basis for the Galilean law. This may at first sight seem all the more surprising as the explanation of acceleration offered in Gassendi's Epistolae de proportione (1646) showed a remarkable resemblance to the ex-

planation given in Mersenne's own Harmonie universelle (1636). We recall that both works attempted to demonstrate, with the help of the double distance rule, that a single force conferring upon a

body a new degree of speed in each moment was capable of augmenting "its speed in the duplicate ratio of the times." And both works-Gassendi's with greater resolve, Mersenne's in a more hypothetical manner-allowed for the possibility that this force was terrestrial attraction.

Moreover, in Mersenne's Novarum observationum tomus III, we find a mathematical analysis of the growth of speeds, times, and

spaces in uniform accelerated motion which bears at first sight a close resemblance to the explanations offered both in Gassendi's

Epistolae de proportione and in Mersenne's own Harmonie univer-

146 Mersenne, Novarum observationum, 135: "[...] vim demonstrationis non habere [...] quemadmodum neque rationes, quae hactenus allatae sunt in gratiam utriusque motus terrae, quidquam demonstrant, ut optime notavit Aristarchi Commentator, etiam si plures vellent eam moveri, ob rationum praestantiam, quae id innuere videntur."




selle.47 But here the similarities end. For while the Mersenne of the Harmonie had believed that his analysis was valid irrespective of whether heavy bodies fell because of some internal principle or because they were attracted by the earth, the old Mersenne explicitly denied the mathematical equivalence of these two hypotheses. In the Novarum observationum tomus III, he argues, against Fabri's and Cazre's impetus theories, that if it could be proven that the motion of heavy bodies was natural, i.e. produced by an internal cause, there would be no reason for not accepting Galileo's law. In the contrary case, however, should it be discovered that bodies are propelled downward by an external force, be it by subtle matter or by terrestrial attraction, one would have to search for another law. 48

The reason why Mersenne had become convinced that it was essential to know the cause responsible for free fall was that there were just too many effects that defied experimental verification. In order to be sure of the validity of the odd-number law, one would in fact have to be able to ascertain: i) that heavy bodies pass through infinite degrees of speed-an assumption whose truth

nobody seems to be able to demonstrate; ii) that the speed of fall increases indefinitely--a notion excluded by adepts of geometry who believe that each body has a maximum velocity beyond which it cannot accelerate; and iii) that the postulate of the inclined planes on which Galileo had founded his entire theory of acceleration, is indeed valid.

When he wrote his Novarum observationum, Mersenne must obviously have abandonned the hope he had expressed in the

Cogitata of 1644 that the Cartesian theory of gravity would yield a mathematical law. The above-mentioned three requirements show, however, that he had completely absorbed the pars destruens of Descartes' thought. We recall in fact the latter's charge that Galileo had founded his science of motion on an undemonstrated

postulate as well as his belief that if the falling body was propelled by an external force acting by contact, its motion had to initiate with a determinate speed and would not have been able to exceed a certain limit.

147 Cf. ibid., 133-134. Here, as already in the Harmonie universelle and in the

Cogitata, Mersenne bases his own explanations on the double distance rule.

"48 Ibid., 136: "Promoveretur illa sententia, si probaretur motum illum gravium esse naturalem; sed cum aliqui contendant esse violentum, sive a principio externo, quod numquam descendant nisi adhibita vi materiae subtilis, ut antea dictum est, vel attractione terrae, non est ex ea parte promota"




We have also seen how in his letters to Mersenne, Descartes had

suggested that the acceleration of fall was not really uniform, since the faster the body was falling the lesser was the additional speed conveyed to it by the subtle matter. Interestingly enough, in the Novarum observationum tomus III, Mersenne seems to accept even this grave objection to Galileo's law. For he writes, albeit without

offering any explanation for it, that the attractive force of the earth would only produce a decreasingly accelerated motion:

Consider also that we still do not know whether heavy bodies are attracted by the earth or rather descend towards it because of some internal urge. If they are attracted, we shall have to speak rather differently about the proportion of their descent than we have done earlier. For [in this case], after having crossed a certain space, they reduce their speed. For the first half of the earth, through which the bodies fall, will strive to pull them down, but will not pull as hard as they approach the center.'49

Mersenne himself cannot say which laws the falling bodies would follow if they were attracted by the earth or pushed by subtle matter. For in both cases, too many variables would come into

play. Nor is he sure whether either of these explanations is to be

accepted. The only thing he knows with certainty is that if either of them were correct, Galileo's law could not be true. The only world in which the odd-number law rules over the fall of bodies is one in which their fall is produced by an inner force. But although this is the hypothesis that appears to have been favored by Galileo, Mersenne finds that he has no reasons for accepting it.

In his discussion of a comparable situation, Richard Westfall has once spoken of the apparent incompatibility between the demands of mathematical mechanics, on the one hand, and those of the mechanical philosophy, on another."15 One feels that Mersenne

149 Mersenne, Novarum observationum, 132: "Adde quod nondum scimus an gravia potius trahantur a terra, quam -ipsa proprio nutu ad eam descendant: quae si trahantur, alio penitus modo de proportione casuum, quam antea dicendum erit: quippe post certum aliquod spatium percursum, suam deinceps velocitatem remittent, cum prima terrae medietas, per quam descendunt gravia, in iisdem retrahendis laboret, vel non ita laboret in trahendis, cum ad centrum accedunt." In a page of the Harmonie universelle, Mersenne refers to the opinion of some people who think that, if falling bodies were attracted by the earth, they would fall "plus viste vers la surface de la terre, que lors qu'il sont plus bas entre la surface & le centre, a raison que la terre entiere les tire vers le centre quand ils tombent par l'air sur sa surface, & qu'elle n'agit plus toute entiere, quand ils descendent sous elle, dautant que toutes les parties qui sont sur les poids, les retirent ' elles tant qu'elles peuvent [...]" (Mersenne, Harmomie universelle, 1: 128). But note that in the Harmonie, Mersenne does not consider this objection to be strong enough to undermine the validity of Galileo's law.

150 R. S. Westfall, Force in Newton's Physics (London, 1971), 47.




was very much aware of this tension. His "ambition to push forward a mathematically certified kind of natural philosophy"''5 was ham-

pered, as it were, by his recognition that the law of acceleration was

incompatible with all strictly mechanistic models of acceleration. Did this recognition result in that famous state of 'mitigated

scepticism' which Richard Popkin has associated with Mersenne, defining it as the awareness that "necessary truths about the nature of reality" are impossible but that "knowledge in a lesser sense, as

convincing or probably truths about appearances" are possible?"52 In one sense, one may answer this question in the positive, for Mersenne continued to believe that Galileo's law, despite all the

uncertainty surrounding it, was, to use Popkin's words, "for all

practical purposes, verifiable and useful."'"5 But in another sense, the story of Mersenne's growing scepticism undermines Popkin's model and its neat distinction between unknown "ultimate causes or real natures" and verifiable "effects and appearances."'54 The belief of the 'mitigated sceptic' that "knowledge of effects is suffi-

cient"'55 does certainly fit Mersenne's early attitude. But we have seen how our Minim gradually came to recognize that the "effects and appearances" were themselves so unclearly perceived, so inex-

actly measured, and thus so poorly understood that it was impossible to analyze them unless one first had an understanding of the "ultimate causes." With respect to the problem surrounding Galileo's law of fall, Mersenne's position must in fact be said to have developed from 'mitigated scepticism' to 'radical scepticism'.

ABSTRACT

This article analyzes the evolution of Mersenne's views concerning the validity of Galileo's theory of acceleration. After publishing, in 1634, a treatise designed to present empirical evidence in favor of Galileo's odd-number law, Mersenne de-

veloped over the years the feeling that only the elaboration of a physical proof could provide sufficient confirmation of its validity. In the present article, I try to show that at the center of Mersenne's worries stood Galileo's assumption that a

falling body had to pass in its acceleration through infinite degrees of speed. His extensive discussions with, or his reading of, Descartes, Gassendi, Baliani, Fabri, Cazre, Deschamps, Le Tenneur, Huygens, and Torricelli led Mersenne to believe that the hypothesis of a passage through infinite degrees of speed was incompatible with any mechanistic explanation of free fall.

•5' Dear, Mersenne, 211. 152 R. H. Popkin, The History of Scepticism from Erasmus to Sponoza (Berkeley,

1979), 129. 153 Ibid., 140. 154 Ibid., 132. 155 Ibid.



Infinite Degrees of Speed: Marin Mersenne and the Debate Over Galileo's Law of Free Fall

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